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0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning " 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title " 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "fixed" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "A q-product tutorial for \+ q-series maple package" }}{PARA 19 "" 0 "" {TEXT -1 12 "Frank Garvan " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 5 "Date:" }{TEXT -1 62 " Tuesday , December 15, 1998. Revised Friday, February 12, 1999" }}{PARA 0 "" 0 "" {TEXT 257 18 "Address of author:" }{TEXT -1 80 " Department of M athematics, University of Florida, Gainesville, Florida 32611" }} {PARA 0 "" 0 "" {TEXT 258 24 "Email address of author:" }{TEXT -1 19 " frank@math.ufl.edu" }}{PARA 0 "" 0 "" {TEXT 259 5 "Note:" }{TEXT -1 73 " This research was supported by the NSF under grant number DMS-9 870052." }}{PARA 0 "" 0 "" {TEXT 260 9 "Keywords:" }{TEXT -1 81 " symb olic computation, products, q-series, theta functions, eta-products, p roduct" }}{PARA 0 "" 0 "" {TEXT -1 23 " identities, partitions" }} {PARA 0 "" 0 "" {TEXT 261 22 "Subject classification" }{TEXT -1 55 ": \+ Primary: 11P81, 68Q40; Secondary: 05A17, 11F20, 33D10" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 8 "Abstract" }}{PARA 0 "" 0 "" {TEXT -1 96 " This is a tutorial for using a new q-series Maple package. The packag e includes facilities for " }}{PARA 0 "" 0 "" {TEXT -1 93 "conversion \+ between q-series and q-products and finding algebraic relations betwee n q-series. " }}{PARA 0 "" 0 "" {TEXT -1 84 " Andrews found an algorit hm for converting a q-series into a product. We provide an " }}{PARA 0 "" 0 "" {TEXT -1 100 "implementation. As an application we are able to effectively find finite q-product factorisations " }}{PARA 0 "" 0 "" {TEXT -1 96 "when they exist thus answering a question of Andrew s. We provide other applications involving " }}{PARA 0 "" 0 "" {TEXT -1 55 " factorisations into theta-functions and eta-products." }}} {EXCHG {PARA 256 "" 0 "" {TEXT 263 67 "Dedicated to George E. Andrews \+ on the occasion of his 60th Birthday" }{TEXT -1 1 " " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 16 "1. Introduction." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "In the study of q-series one is quite often interested i n identifying generating functions as infinite " }}{PARA 0 "" 0 "" {TEXT -1 65 "products. The classic example is the Rogers-Ramanujan id entity: " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&)%\"qG *$)%\"nG\"\"#\"\"\"F.&-%!G6$F)F)6#F,!\"\"/F,;\"\"!%)infinityG-%(Produc tG6$*&F.F.*&,&\"\"\"F?)F),&F,\"\"&!\"\"F?FC\"\"\",&F?F?)F),&F,FB!\"%F? FC\"\"\"F4/F,;F?F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Here we have used the notation in (2.2). It can be shown that the left-side of this identity is the" } }{PARA 0 "" 0 "" {TEXT -1 107 "generating function for partitions whos e parts differ by at least two. The identity is equivalent to saying" }}{PARA 0 "" 0 "" {TEXT -1 95 "the number of such partitions of n is e quinumerous with partitions of n into parts congruent to" }}{PARA 0 " " 0 "" {TEXT -1 14 "+/-1 (mod 5).." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The main goals of the p ackage are to provide facility for handling the following problems." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "1. Conversion of a given q-se ries into an ``infinite'' product." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "2. Factorization of a given rational function into a finite q -product if one exists." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "3. F ind algebraic relations (if they exist) among the q-series in a given \+ list." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "A q-product has the form" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%(ProductG6$),&\"\"\"F()%\"qG%\"jG!\"\"&%\"bG6#F+/F+;F (%\"NG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "where the " }{XPPEDIT 18 0 "b[j];" "6#&%\"bG6 #%\"jG" }{TEXT -1 17 " are integers." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "In [4, sectio n 10.7], George Andrews also considered Problems 1 and 2, and asked fo r an " }}{PARA 0 "" 0 "" {TEXT -1 84 "easily accessible implementatio n. We provide implementations as well as considering" }}{PARA 0 "" 0 " " {TEXT -1 99 "factorisations into theta-products and eta-products. Th e package provides some basic functions for " }}{PARA 0 "" 0 "" {TEXT -1 101 "computing q-series expansions of eta functions, theta function s, Gaussian polynomials and q-products." }}{PARA 0 "" 0 "" {TEXT -1 105 "It also has a function for sifting out coefficients of a q-series . It also has the basic infinite product" }}{PARA 0 "" 0 "" {TEXT -1 96 "identities: the triple product identity, the quintuple product ide ntity and Winquist's identity." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 30 "1.1 Installation instructions ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 0 7 "qseries" } {TEXT -1 84 ". package can be downloaded via the WWW. First use your \+ favorite browser to access " }}{PARA 0 "" 0 "" {TEXT -1 8 "the URL:" } }{PARA 257 "" 0 "" {TEXT -1 44 "http://www.math.ufl.edu/simfrank/qmapl e.html" }}{PARA 0 "" 0 "" {TEXT -1 98 "then follow the directions on t hat page. There are two versions: one for UNIX and one for WINDOWS." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 18 "2. Basic functions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "We describe the basic functi ons in the package which are used to build q-series." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 22 "2.1. Finite q-products" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 25 "2.1.1. Rising q-factorial" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 13 "aqprod(a,q,n)" }{TEXT -1 21 " returns the product" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "(2.2)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%!G6$%\"aG%\"qG6#%\"nG**,&\"\"\"F.F(!\"\"F.,&F.F.*&F(F.F)F.F/ F.%$...GF.,&F.F.*&F(\"\"\")F),&F+F.F/F.F.F/F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "We also use the notation" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%!G6$%\"aG%\"qG6#%)infinityG-%(ProductG6$,&\"\"\"F0* &F(F0)F),&%\"nGF0!\"\"F0F0F5/F4;F0F+" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 26 "2.1.2 Gaussian polynomials" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "0 <= m;" "6#1\"\"!%\"mG" }{XPPEDIT 18 0 "`` <= n;" "6#1%!G%\"nG" }{TEXT -1 4 ", " }{TEXT 0 11 "qbin(q,m,n)" } {TEXT -1 62 " returns the Gaussian polynomial (or q-binomial coeffic ient)" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7$7#%\"n G7#%\"mG*&&-%!G6#%\"qG6#F)\"\"\"*&&F.6#F+\"\"\"&F.6#,&F)\"\"\"F+!\"\" \"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "otherwise it retur ns 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 21 "2.2 Infinite products" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 27 "2.2.1 Dedekind eta products" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Suppose " }{XPPEDIT 18 0 "0 < Re(tau);" "6#2\"\"!-%#ReG 6#%$tauG" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "q = ` exp`(2*Pi*i*tau) ;" "6#/%\"qG-%%~expG6#**\"\"#\"\"\"%#PiGF*%\"iGF*%$tauGF*" }{TEXT -1 53 ". The Dedekind eta function [27, p.121] is defined by" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$etaG6#%$tauG*&-%%~expG6#,$*(%#PiG \"\"\"%\"iGF/F'F/#F/\"#7F/-%(ProductG6$,&F/F/-F*6#,$**F.\"\"\"F0F;%\"n GF/F'F;\"\"#!\"\"/F<;F/%)infinityGF/" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&)%\"qG#\"\"\"\"#C\"\"\"-%(ProductG6$,&F)F))F'%\"n G!\"\"/F1;F)%)infinityGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 12 ".etaq (q,k,T)" }{TEXT -1 40 " returns the q-series expansion (up to " } {XPPEDIT 18 0 "q^T;" "6#)%\"qG%\"TG" }{TEXT -1 20 ") of the eta produc t" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%(ProductG6$,&\"\"\"F')% \"qG*&%\"kGF'%\"nGF'!\"\"/F,;F'%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "This corresponds to the eta function " }{XPPEDIT 18 0 "e ta(k*tau);" "6#-%$etaG6#*&%\"kG\"\"\"%$tauGF(" }{TEXT -1 60 " except \+ for a power of q. Eta products occur frequently in " }}{PARA 0 "" 0 " " {TEXT -1 105 "the study of q-series. For example, the generating fun ction for p(n), the number of partitions of n, can " }}{PARA 0 "" 0 " " {TEXT -1 13 "be written as" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$SumG6$*&-%\"pG6#%\"nG\"\"\")%\"qGF+F,/F+;\"\"!%)infinityG-%(Prod uctG6$*&\"\"\"F7,&F,F,F-!\"\"!\"\"/F+;F,F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "See [1, pp. 3-4]. The generating function for the numbe r of partitions of " }{TEXT 264 1 "n" }{TEXT -1 10 " that are " } {TEXT 265 1 "p" }{TEXT -1 14 "-cores [19], " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "a[p](n);" "6#-&%\"aG6#%\"pG6#%\"nG" }{TEXT -1 16 ", can be written" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&-&% \"aG6#%\"pG6#%\"nG\"\"\")%\"qGF.F//F.;\"\"!%)infinityG-%(ProductG6$*&) ,&F/F/)F1*&F,F/F.F/!\"\"F,\"\"\",&F/F/F0F>!\"\"/F.;F/F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Recently, Granville and Ono [21] were a ble to prove a long-standing conjecture in group " }}{PARA 0 "" 0 "" {TEXT -1 98 "representation theory using elementary and function-theor etic properties of the eta product above." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 22 "2.2.2. Theta func tions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Jacobi [24, Vol I, pp. \+ 497--538] defined four theta functions " }{XPPEDIT 18 0 "theta[i](z, q);" "6#-&%&thetaG6#%\"iG6$%\"zG%\"qG" }{TEXT -1 12 ", i=1,2,3,4." }} {PARA 0 "" 0 "" {TEXT -1 100 "See also [41, Ch. XXI]. Each theta fun ction can be written in terms of the others using a simple " }}{PARA 0 "" 0 "" {TEXT -1 60 "change of variables. For this reason, it is com mon to define" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&thetaG6$% \"zG%\"qG-%$SumG6$*&)F'%\"nG\"\"\")F(*$)F.\"\"#\"\"\"F//F.;,$%)infinit yG!\"\"F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 12 "theta(z,q,T)" }{TEXT -1 55 " returns the truncated theta-series " }}} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)%\"zG%\"iG\"\"\")% \"qG*$)F)\"\"#\"\"\"F*/F);,$%\"TG!\"\"F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The case z=1 of Jacobi's theta functions occurs quite fre quently. We define" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%&the taG6#\"\"#6#%\"qG-%$SumG6$)F*,$*$),&%\"nG\"\"\"F4F4F(\"\"\"#F4F(/F3;,$ %)infinityG!\"\"F:" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%&the taG6#\"\"$6#%\"qG-%$SumG6$)F**$)%\"nG\"\"#\"\"\"/F1;,$%)infinityG!\"\" F7" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%&thetaG6#\"\"%6#%\"q G-%$SumG6$*&)!\"\"%\"nG\"\"\")F**$)F1\"\"#\"\"\"F2/F1;,$%)infinityGF0F ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 11 "theta2(q,T)" }{TEXT -1 3 ", \+ " }{TEXT 0 11 "theta3(q,T)" }{TEXT -1 3 ", " }{TEXT 0 11 "theta4(q,T) " }{TEXT -1 42 " (resp.) returns the q-series expansion to" }}{PARA 0 "" 0 "" {TEXT -1 7 "order " }{XPPEDIT 18 0 "O(q^T);" "6#-%\"OG6#)%\"q G%\"TG" }{TEXT -1 5 " of " }{XPPEDIT 18 0 "theta[2](q);" "6#-&%&theta G6#\"\"#6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "theta[3](q);" "6#-& %&thetaG6#\"\"$6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "theta[4](q); " "6#-&%&thetaG6#\"\"%6#%\"qG" }{TEXT -1 17 " respectively. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 266 1 "a" }{TEXT -1 6 ", and " }{TEXT 267 1 "b" }{TEXT -1 35 " be positive integers and supp ose |" }{TEXT 268 1 "q" }{TEXT -1 36 "|<1. Infinite products of the \+ form" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#*&&-%!G6$)%\"qG%\"aG)F )%\"bG6#%)infinityG\"\"\"&-F&6$)F),&F,F/F*!\"\"F+F-F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "occur quite frequently in the theory of \+ partitions and q-series. For example the right side of the " }}{PARA 0 "" 0 "" {TEXT -1 76 "Rogers-Ramanujan identity is the reciprocal of \+ the product with (a,b)=(1,5)." }}{PARA 0 "" 0 "" {TEXT -1 40 "In (3.4 ) we will see how the function " }{TEXT 0 11 "jacprodmake" }{TEXT -1 40 " can be used to identify such products." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 22 "3. Product Conversion." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "In [1, p. 233], [3, section 10.7] there is a very nice and useful algorithm for converting a q-series into " }}{PARA 0 "" 0 "" {TEXT -1 87 "an in finite product. Any given q-series may be written formally as an infi nite product" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"\"F%-%$ SumG6$*&&%\"bG6#%\"nGF%)%\"qGF-F%/F-;F%%)infinityGF%-%(ProductG6$),&F% F%F.!\"\",$&%\"aGF,F8F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Here w e assume the " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT -1 81 " are integers. By taking the logarithmic derivative of both side s we can obtain" }}{PARA 0 "" 0 "" {TEXT -1 15 " the recurrence" }}} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"nG\"\"\"&%\"bG6#F%F&-%$Su mG6$*&&F(6#,&F%F&%\"jG!\"\"F&-F+6$*&%\"dGF&&%\"aG6#F6F&/F6;%-divisor~o f~jG%!GF&/F1;F&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Letting " } {XPPEDIT 18 0 "a[n] = 1;" "6#/&%\"aG6#%\"nG\"\"\"" }{TEXT -1 40 " we obtain the well-known special case" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"nG\"\"\"-%\"pG6#F%F&-%$SumG6$*&-F(6#,&F%F&%\"jG! \"\"F&-%&sigmaG6#F1F&/F1;F&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 " We can also easily construct a recurrence for the " }{XPPEDIT 18 0 "a [n];" "6#&%\"aG6#%\"nG" }{TEXT -1 28 " from the recurrence above." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The function " }{TEXT 0 8 "prod make" }{TEXT -1 75 " is an implementation of Andrews' algorithm. Ot her related functions are" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 0 7 "etamake" }{TEXT -1 9 " and " }{TEXT 0 11 "jacprodmake" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 12 "3.1 prodmake" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 15 " prodmake(f,q,T)" }{TEXT -1 25 " converts the q-series " }{TEXT 269 1 "f" }{TEXT -1 43 " into an infinite product that agrees with " } {TEXT 270 1 "f" }{TEXT -1 4 " to " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "O( q^T);" "6#-%\"OG6#)%\"qG%\"TG" }{TEXT -1 72 ". Let's take a look at \+ the left side of the Rogers-Ramanujan identity." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " wit h(qseries): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " x:=1: " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " for n from 1 to 8 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " x := x + q^(n*n)/aqprod(q,q,n):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " x := series(x,q,50);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " prodmake(x,q,40);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"xG+cq%\"qG\"\"\"\"\"!F'\"\"\"F'\"\"#F'\"\"$\"\"#\"\"%F,\"\"&\" \"$\"\"'F/\"\"(\"\"%\"\")\"\"&\"\"*\"\"'\"#5\"\"(\"#6\"\"*\"#7\"#5\"#8 \"#7\"#9\"#9\"#:\"#<\"#;\"#>\"#<\"#B\"#=\"#E\"#>\"#J\"#?\"#N\"#@\"#T\" #A\"#Y\"#B\"#a\"#C\"#h\"#D\"#q\"#E\"#z\"#F\"#\"*\"#G\"$-\"\"#H\"$<\"\" #I\"$J\"\"#J\"$\\\"\"#K\"$n\"\"#L\"$*=\"#M\"$6#\"#N\"$R#\"#O\"$m#\"#P \"$*H\"#Q\"$L$\"#R\"$u$\"#S\"$:%\"#T\"$l%\"#U\"$:&\"#V\"$v&\"#W\"$P'\" #X\"$4(\"#Y\"$$y\"#Z\"$r)\"#[\"$h*\"#\\-%\"OG6#F'\"#]" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*B,&\"\"\"F'%\"qG!\"\"\"\"\",&F'F'*$)F( \"\"%F$F)\"\"\",&F'F'*$)F(\"\"'F$F)\"\"\",&F'F'*$)F(\"\"*F$F)\"\"\",&F 'F'*$)F(\"#6F$F)\"\"\",&F'F'*$)F(\"#9F$F)\"\"\",&F'F'*$)F(\"#;F$F)\"\" \",&F'F'*$)F(\"#>F$F)\"\"\",&F'F'*$)F(\"#@F$F)\"\"\",&F'F'*$)F(\"#CF$F )\"\"\",&F'F'*$)F(\"#EF$F)\"\"\",&F'F'*$)F(\"#HF$F)\"\"\",&F'F'*$)F(\" #JF$F)\"\"\",&F'F'*$)F(\"#MF$F)\"\"\",&F'F'*$)F(\"#OF$F)\"\"\",&F'F'*$ )F(\"#RF$F)\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "We have rediscovered the right \+ side of the Rogers-Ramanujan identity!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 81 "________________ _________________________________________________________________" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 88 "______________________________________________________________ __________________________" }}{PARA 0 "" 0 "" {TEXT 271 10 "Exercise 1 " }{TEXT -1 52 ". Find (and prove) a product form for the q-series" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)%\"qG*$)%\"nG\"\"#\"\"\" F-&-%!G6$F(F(6#,$F+F,!\"\"/F+;\"\"!%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 109 "The identity you find is originally due to Rogers [34, p .330]. See also Andrews [2, pp.38--39] for a list of" }}{PARA 0 "" 0 "" {TEXT -1 20 "some related papers." }}{PARA 0 "" 0 "" {TEXT 273 88 " ______________________________________________________________________ __________________" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 12 "3.2 qfactor ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The function " }{TEXT 0 7 " qfactor" }{TEXT -1 19 " is a version of " }{TEXT 0 8 "prodmake" } {TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 12 "qfactor(f,T)" } {TEXT -1 38 "attempts to write a rational function " }{TEXT 275 1 "f" }{TEXT -1 4 " in " }{TEXT 276 1 "q" }{TEXT -1 52 " as a q-product, ,ie ., as a product of terms of the " }}{PARA 0 "" 0 "" {TEXT -1 6 "form ( " }{XPPEDIT 18 0 "1-q^i;" "6#,&\"\"\"\"\"\")%\"qG%\"iG!\"\"" }{TEXT -1 24 "). The second argument " }{TEXT 277 2 "T " }{TEXT -1 66 "is op tional. It specifies an an upper bound for the exponents of " }{TEXT 278 1 "q" }{TEXT -1 8 " that " }}{PARA 0 "" 0 "" {TEXT -1 30 "can oc cur in the product. If " }{TEXT 279 1 "T" }{TEXT -1 51 " is not speci fied it is given a default value of 4" }{TEXT 281 1 "d" }{TEXT -1 9 " +3 where " }{TEXT 280 1 "d" }{TEXT -1 29 " is the maximum of the degre e" }}{PARA 0 "" 0 "" {TEXT -1 3 "in " }{TEXT 282 1 "q" }{TEXT -1 108 " of the numerator and denominator. The algorithm is quite simple. Fi rst the function is factored as usual," }}{PARA 0 "" 0 "" {TEXT -1 18 "and then it uses " }{TEXT 0 8 "prodmake" }{TEXT -1 100 " to do furth er factorisation into q-products. Thus even if only part of the functi on can be written " }}{PARA 0 "" 0 "" {TEXT -1 16 "as a q-product " } {TEXT 0 8 "qfactor " }{TEXT -1 20 " is able to find it." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "A s an example we consider some rational functions " }{XPPEDIT 18 0 "T( r,h);" "6#-%\"TG6$%\"rG%\"hG" }{TEXT -1 46 " introduced by Andrews [ 4, p.14] to explain" }}{PARA 0 "" 0 "" {TEXT -1 67 "Rogers's [34] fi rst proof of the Rogers-Ramanujan identities. The " }{XPPEDIT 18 0 "T( r,h);" "6#-%\"TG6$%\"rG%\"hG" }{TEXT -1 37 " are defined recursively \+ as follows:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "(3.3) " } {XPPEDIT 18 0 "T(r,0) = 1;" "6#/-%\"TG6$%\"rG\"\"!\"\"\"" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "(3.4) " }{XPPEDIT 18 0 " T(r,1) = 0;" "6#/-%\"TG6$%\"rG\"\"\"\"\"!" }{TEXT -1 3 ", ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "(3.5) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6$%\"rG%\"NG,$-%$SumG6$*&-%'matrixG6#7$7#,&F'\" \"\"%\"jG\"\"#7#F5F4-F%6$F3,&F(F4F5!\"#F4/%#2jG;F4F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(qseries):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 " T:=proc(r,j)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " option remember; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " \+ local x,k; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " x:=0; " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " if j=0 or j=1 then" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 23 " RETURN((j-1)^2): " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 9 " else " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " \+ for k from 1 to floor(j/2) do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " x:=x-qbin(q,k,r+2*k)*T(r+2*k,j-2*k); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " RETURN(expand(x)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " \+ fi: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " t8:=T(8,8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#t8G,fo*$)%\"qG\"#I\"\"\"\"#G*$)F(\"#9F*\"#:*$)F(\"#5 F*\"\"&*$)F(\"#CF*\"#Q*$)F(\"#EF*\"#P*$)F(F/F*\"#<*$)F(\"\"(F*\"\"\"*$ )F(\"#7F*\"\"**$)F(\"#AF*F;*$)F(\"#NF*\"#6*$)F(\"\"'F*FB*$)F(\"\")F*\" \"#*$)F(FFF*\"\"$*$)F(FMF*FP*$)F(\"#8F*FM*$)F(\"#;F*\"#@*$)F(F>F*\"#B* $)F(\"#=F*F+*$)F(\"#>F*\"#H*$)F(\"#?F*\"#L*$)F(FjnF*\"#M*$)F(F]oF*\"#O *$)F(\"#DF*F^p*$)F(\"#FF*F[p*$)F(F+F*Fho*$)F(FdoF*Fdo*$)F(\"#JF*F]o*$) F(\"#KF*Fjn*$)F(FhoF*F>*$)F(F[pF*F/*$)F(F^pF*FF*$)F(F;F*FP*$)F(F7F*F3* $)F(\"#RF*FW*$)F(\"#SF*FT*$)F(\"#TF*FB*$)F(\"#UF*FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " factor(t8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*0)%\"qG\"\"'\"\"\",,*$)F%\"\"%F'\"\"\"*$)F%\"\"$F'F,*$ )F%\"\"#F'F,F%F,F,F,F,,,F)F,F-!\"\"F0F,F%F4F,F,F,,8*$)F%\"#5F'F,*$)F% \"\"*F'F,*$)F%\"\")F'F,*$)F%\"\"(F'F,*$F$F'F,*$)F%\"\"&F'F,F)F,F-F,F0F ,F%F,F,F,F,,&F)F,F,F,F,,(FBF,F-F,F,F,F,,&F " 0 "" {MPLTEXT 1 0 18 " qfactor(t8,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*,)%\"qG\"\"'\"\"\",&\"\"\"F**$)F&\"#5F(!\"\"F*,&F*F* *$)F&\"#6F(F.F*,&F*F**$)F&\"\"*F(F.F*,&F*F**$)F&\"#;F(F.F*F(**,&F*F*F& F.\"\"\",&F*F**$)F&\"\"#F(F.\"\"\",&F*F**$)F&\"\"%F(F.\"\"\",&F*F**$)F &\"\"$F(F.\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Observe \+ how we used " }{TEXT 0 6 "factor" }{TEXT -1 12 " to factor " }{TEXT 283 2 "t8" }{TEXT -1 40 " into cyclotomic polynomials. However, " } {TEXT 0 7 "qfactor" }{TEXT -1 4 " was" }}{PARA 0 "" 0 "" {TEXT -1 16 " able to factor " }{TEXT 284 2 "t8" }{TEXT -1 16 " as a q-product." }} {PARA 0 "" 0 "" {TEXT -1 11 "We see that" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#~TG6$\"\")F'*&*(&-%!G6$*$)%\"qG\"\"*\"\"\"F06#\"\"$ \"\"\",&F5F5*$)F0\"#;F2!\"\"F5)F0\"\"'F2F2&-F,6$F0F06#\"\"%!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 81 "______________________________________________________________ ___________________" }}{PARA 0 "" 0 "" {TEXT 287 13 "EXERCISE 2. " } {TEXT -1 6 " Use " }{TEXT 0 7 "qfactor" }{TEXT -1 17 " to factorize \+ T(" }{TEXT 288 3 "r,n" }{TEXT -1 26 ") for different values of " } {TEXT 289 1 "r" }{TEXT -1 6 " and " }{TEXT 290 1 "n" }{TEXT -1 13 ". \+ Then write " }}{PARA 0 "" 0 "" {TEXT -1 2 "T(" }{TEXT 291 1 "r" } {TEXT -1 1 "," }{TEXT 292 1 "n" }{TEXT -1 45 ") (defined above) as a q -product for general " }{TEXT 293 1 "r" }{TEXT -1 5 " and " }{TEXT 294 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 286 81 "____________ _____________________________________________________________________ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "For our next example we examine the sum " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,)!\"\"%\"kG\"\"\")%\"qG,$*& F)F*,&F)\"\"$F*F*F*#F*\"\"#F*-%'matrixG6#7$7#,&%\"bGF*%\"cGF*7#,&F:F*F )F*F*-F46#7$7#,&F:F*%\"aGF*7#,&FBF*F)F*F*-F46#7$7#,&FBF*F9F*7#,&F9F*F) F*F*/F);,$%)infinityGF(FO" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " dixson:=proc(a,b,c,q) " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " local x,k,y; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " x:=0: y:=min(a,b,c): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " for k from -y to y do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " x:=x+(-1)^(k)*q^(k*(3*k+1)/2)* " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " qbin(q,c+k,b+c)*qbin(q,a+k,c+a)*qbi n(q,b+k,a+b); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " od: " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " RETURN(x): " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 6 " end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " dx := expand(dixson(5,5,5,q)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " qfactor(dx,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&*6,&\"\"\"F&*$)%\"qG\"#5\"\"\"!\"\"F&,&F&F&*$)F)\"\"(F+F,F&,&F&F&*$ )F)\"#9F+F,F&,&F&F&*$)F)\"#6F+F,F&,&F&F&*$)F)\"\"'F+F,F&,&F&F&*$)F)\"# 7F+F,F&,&F&F&*$)F)\"#:F+F,F&,&F&F&*$)F)\"#8F+F,F&,&F&F&*$)F)\"\"*F+F,F &,&F&F&*$)F)\"\")F+F,F&F+*,),&F&F&F)F,\"\"#F+),&F&F&*$)F)\"\"#F+F,\"\" #F+),&F&F&*$)F)\"\"&F+F,\"\"#F+),&F&F&*$)F)\"\"%F+F,\"\"#F+),&F&F&*$)F )\"\"$F+F,\"\"#F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "We find that " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "sum_.k=-infty.^infty (- 1)^k q^.k (3 k+1)/2." }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Su mG6$*()!\"\"%\"kG\"\"\")%\"qG,$*&F*F+,&F*\"\"$F+F+F+#F+\"\"#F+)-%'matr ixG6#7$7#\"#57#,&\"\"&F+F*F+F1\"\"\"/F*;,$%)infinityGF)FB*&&-%!G6$*$)F -\"\"'F>F-6#F:F>*$)&-FF6$F-F-6#F=\"\"#F>!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 81 "_____________________________________________________ ____________________________" }}{PARA 0 "" 0 "" {TEXT 296 11 "EXERCISE 3." }{TEXT -1 19 " Write the sum " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*()!\"\"%\"kG\"\"\")%\"qG,$*&F)F*,&F)\"\"$F*F*F*#F*\"\" #F*)-%'matrixG6#7$7#,$%\"aGF27#,&F:F*F)F*F0\"\"\"/F);,$%)infinityGF(FA " }}{PARA 0 "" 0 "" {TEXT -1 36 "as a q-product for general integral \+ " }{TEXT 297 1 "a" }{TEXT -1 68 ". The identity you obtain is a specia l case of [4, Eq.(4.24), p.38]." }}{PARA 0 "" 0 "" {TEXT 298 81 "_____ ______________________________________________________________________ ______" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 12 "3.3 etamake" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Recall from (2.2.1) that " } {TEXT 0 4 "etaq" }{TEXT -1 69 " is the function to use for computing \+ q-expansions of eta products. " }}{PARA 0 "" 0 "" {TEXT -1 101 "If one wants to apply the theory of modular forms to q-series it is quite us eful to determine whether" }}{PARA 0 "" 0 "" {TEXT -1 101 "a given q-s eries is a product of eta functions. The function in the package for d oing this conversion" }}{PARA 0 "" 0 "" {TEXT -1 4 "is " }{TEXT 0 7 " etamake" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 15 "etamake (f,q,T) " }{TEXT -1 30 "will write the given q-series " }{TEXT 300 1 " f" }{TEXT -1 49 " as a product of eta functions which agrees with " }} {PARA 0 "" 0 "" {TEXT 301 1 "f" }{TEXT -1 9 " up to " }{XPPEDIT 18 0 "q^T;" "6#)%\"qG%\"TG" }{TEXT -1 65 ". As an example, let's see ho w we can write the theta functions" }}{PARA 0 "" 0 "" {TEXT -1 16 "as \+ eta products." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "t2:=theta2 (q,100)/q^(1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G,<*$)%\"qG\" $K\"\"\"\"\"\"#*$)F(\"$5\"F*F+*$)F(\"#!*F*F+*$)F(\"#sF*F+*$)F(\"#cF*F+ *$)F(\"#UF*F+*$)F(\"#IF*F+*$)F(\"#?F*F+*$)F(\"#7F*F+*$)F(\"\"'F*F+*$)F (F+F*F+F+\"\"\"*$)F(\"$c\"F*FI" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "etamake(t2,q,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$)- %$etaG6#,$%$tauG\"\"%\"\"#\"\"\"F.*&)%\"qG#\"\"\"F,F.-F(6#,$F+F-\"\"\" !\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "t3:=theta3(q,100 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G,:*$)%\"qG\"$@\"\"\"\"\"\" #*$)F(\"$+\"F*F+*$)F(\"#\")F*F+*$)F(\"#kF*F+*$)F(\"#\\F*F+*$)F(\"#OF*F +*$)F(\"#DF*F+*$)F(\"#;F*F+*$)F(\"\"*F*F+*$)F(\"\"%F*F+F(F+\"\"\"FG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "etamake(t3,q,100);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&*$)-%$etaG6#,$%$tauG\"\"#\"\"&\"\"\" F-*&)-F'6#,$F*\"\"%\"\"#F-)-F'6#F*\"\"#F-!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "t4:=theta4(q,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t4G,:*$)%\"qG\"$@\"\"\"\"!\"#*$)F(\"$+\"F*\"\"#*$)F( \"#\")F*F+*$)F(\"#kF*F/*$)F(\"#\\F*F+*$)F(\"#OF*F/*$)F(\"#DF*F+*$)F(\" #;F*F/*$)F(\"\"*F*F+*$)F(\"\"%F*F/F(F+\"\"\"FH" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "etamake(t4,q,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*$)-%$etaG6#%$tauG\"\"#\"\"\"F+-F'6#,$F)F*!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "We are led to the well-known ident ities:" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%&thetaG\"\"\"-&% !G6#\"\"#6#%\"qGF&,$*&*$)-%$etaG6#,$%$tauG\"\"%F+\"\"\"F8*&)F-#\"\"\"F 7F8-F36#,$F6F+\"\"\"!\"\"F+" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&%&thetaG\"\"\"-&%!G6#\"\"$6#%\"qGF&*&*$)-%$etaG6#,$%$tauG\"\"#\"\" &\"\"\"F8*&)-F26#,$F5\"\"%\"\"#F8)-F26#F5\"\"#F8!\"\"" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%&thetaG\"\"\"-&%!G6#\"\"%6#%\"qGF& *&*$)-%$etaG6#%$tauG\"\"#\"\"\"F6-F26#,$F4F5!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 60 "The idea of the algorithm is quite simple. Given a q-series " }{TEXT 302 1 "f" }{TEXT -1 60 " (say with leading coeffici ent 1) one just keeps recursively" }}{PARA 0 "" 0 "" {TEXT -1 117 "mul tiplying by powers of the right eta function until the desired term s agree. For example, suppose we are given a" }}{PARA 0 "" 0 "" {TEXT -1 9 " q-series" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"fG,(\"\"\"F&*&% \"cGF&)%\"qG%\"kGF&F&%$...GF&" }}{PARA 0 "" 0 "" {TEXT -1 39 "Then the next step is to multiply by " }{TEXT 0 16 "etaq(q,k,T)^(-c)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 305 81 "________________________________________ _________________________________________" }}{PARA 0 "" 0 "" {TEXT 303 11 "EXERCISE 4." }}{PARA 0 "" 0 "" {TEXT -1 19 "Define the q-serie s" }}{PARA 0 "" 0 "" {TEXT -1 8 "where " }{XPPEDIT 18 0 "omega = ` e xp`(2*Pi*i/3);" "6#/%&omegaG-%%~expG6#**\"\"#\"\"\"%#PiGF*%\"iGF*\"\"$ !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 64 "Two of the three functions above can be written as eta products." }}{PARA 0 "" 0 "" {TEXT -1 19 "Can you find them? " }}{PARA 0 "" 0 "" {TEXT 304 4 "Hint " }{TEXT -1 28 ": It would be wise to define" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "omega:=RootOf(z^2+z+1=0);" }}{PARA 0 "" 0 "" {TEXT -1 39 "See [12] for the answer and much more." }}{PARA 0 "" 0 "" {TEXT 306 81 "________________________________________________________ _________________________" }}{PARA 8 "" 1 "" {TEXT -1 48 "Error, (in R ootOf) expression independent of, _Z" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 16 "3.4 jacprodmake" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "In (2 .2.2) we observed that the right side of the Rogers-Ramanujan identit y could be written in" }}{PARA 0 "" 0 "" {TEXT -1 42 " terms of a Jaco bi product. The function " }{TEXT 0 12 "jacprodmake " }{TEXT -1 39 "c onverts a q-series into a Jacobi-type " }}{PARA 0 "" 0 "" {TEXT -1 41 "product if one exists. Given a q-series " }{TEXT 307 1 "f" }{TEXT -1 4 ", j" }{TEXT 0 17 "acprodmake(f,q,T)" }{TEXT -1 22 " attempts to convert " }{TEXT 308 1 "f" }{TEXT -1 8 " into a " }}{PARA 0 "" 0 "" {TEXT -1 45 "product of theta functions that agrees with " }{TEXT 309 1 "f" }{TEXT -1 12 " to order " }{XPPEDIT 18 0 "O(q^T);" "6#-%\" OG6#)%\"qG%\"TG" }{TEXT -1 36 ". Each theta-function has the form" } }{PARA 0 "" 0 "" {TEXT 0 15 "JAC.(a,b,infty)" }{TEXT -1 9 ", where " }{TEXT 310 1 "a" }{TEXT -1 2 ", " }{TEXT 311 1 "b" }{TEXT -1 19 " are \+ integers and " }{XPPEDIT 18 0 "0 <= a;" "6#1\"\"!%\"aG" }{XPPEDIT 18 0 "`` < b;" "6#2%!G%\"bG" }{TEXT -1 8 ". If " }{XPPEDIT 18 0 "0 < a ;" "6#2\"\"!%\"aG" }{TEXT -1 7 ", then " }{TEXT 0 16 "JAC.(a,b,infty) \+ " }}{PARA 0 "" 0 "" {TEXT -1 33 "corresponds to the theta-product" }} }{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#*(&-%!G6$)%\"qG%\"aG)F)%\"bG6# %)infinityG\"\"\"&-F&6$)F),&F,F/F*!\"\"F+F-F/&-F&6$F+F+F-F/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "We call this a theta product because it is " }{XPPEDIT 18 0 "th eta(-q^((b-2*a)/2),q^(b/2));" "6#-%&thetaG6$,$)%\"qG*&,&%\"bG\"\"\"*& \"\"#F,%\"aGF,!\"\"F,\"\"#F0F0)F(*&F+F,\"\"#F0" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "The " }{TEXT 0 13 "jacprodmake \+ " }{TEXT -1 34 " function is really a variant of " }{TEXT 0 11 "prodm ake. " }{TEXT -1 18 "It involves using " }}{PARA 0 "" 0 "" {TEXT 0 9 "prodmake " }{TEXT -1 73 " to compute the sequence of exponents and th en searching for periodicity." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 " If " }{XPPEDIT 18 0 "a = \+ 0;" "6#/%\"aG\"\"!" }{TEXT -1 7 ", then " }{TEXT 0 15 "JAC.(0,b,infty) " }{TEXT -1 32 " corresponds to the eta-product" }}}{EXCHG {PARA 11 " " 1 "" {XPPMATH 20 "6#&-%!G6$)%\"qG%\"bGF'6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "We note that this product can also be tho ught of as a theta-product" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "sin ce can be written" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%!G6$) %\"qG%\"bGF(6#%)infinityG*(&-F&6$F()F),$F*\"\"$F+\"\"\"&-F&6$)F),$F*\" \"#F1F+F4&-F&6$F1F1F+F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Let's \+ re-examine the Rogers-Ramanujan identity." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 " with(qseries): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " x:=1: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " for n from 1 to 8 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " \+ x:=x+q^(n*n)/aqprod(q,q,n): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " \+ od: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " x:=series(x,q, 50): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " y:=jacprodmake( x,q,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG*&-%$JACG6%\"\"!\"\" &%)infinityG\"\"\"-F'6%\"\"\"F*F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " z:=jac2prod(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"zG*&\"\"\"F&*&&-%\"~G6$%\"qG*$)F,\"\"&F&6#%)infinityG\"\"\"&-F*6$ *$)F,\"\"%F&F-F0\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "No te that we were able to observe that the left side of the Rogers-Raman ujan identity (at least up " }}{PARA 0 "" 0 "" {TEXT -1 8 "through " } {XPPEDIT 18 0 "q^40;" "6#*$%\"qG\"#S" }{TEXT -1 73 ") can be written a s a quotient of theta functions. We used the function " }{TEXT 0 8 "j ac2prod" }{TEXT -1 5 ", to " }}{PARA 0 "" 0 "" {TEXT -1 77 "simplify t he result and get it into a more recognizable form. The function " } {TEXT 0 17 "jac2prod(jacexpr)" }}{PARA 0 "" 0 "" {TEXT -1 102 "convert s a product of theta functions into q-product form; ie., as a product \+ of functions of the form" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "``(q^a,q^b )[infinity];" "6#&-%!G6$)%\"qG%\"aG)F(%\"bG6#%)infinityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Here " }{TEXT 0 7 "jacexpr " }{TEXT -1 39 " is a product (or quotient) of terms " }{TEXT 0 15 " JAC(i,j, infty)" }{TEXT -1 8 ", where " }{TEXT 312 1 "i" }{TEXT -1 2 " , " }{TEXT 313 1 "j" }{TEXT -1 13 " are integers" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 5 "and " }{XPPEDIT 18 0 "0 <= i;" "6#1\"\"!%\"iG" } {XPPEDIT 18 0 "`` < j;" "6#2%!G%\"jG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "A related function is " } {TEXT 0 10 "jac2series" }{TEXT -1 61 ". This converts a Jacobi-type pr oduct into a form better for " }}{PARA 0 "" 0 "" {TEXT -1 92 "computin g its q-series. It simply replaces each Jacobi-type product with its \+ corresponding " }}{PARA 0 "" 0 "" {TEXT -1 13 "theta-series." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " with(qseries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " x:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " for n from 0 to 10 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " x := x + q^(n*(n+1)/2)*aqprod(-q,q,n)/aqprod(q,q,2 *n+1):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " od: " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 " x:=series(x,q,50):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " jp:=jacprodmake(x,q,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#jpG*&*$)-%$JACG6%\"\"!\"#9%)infinityG\"\"'\"\"\"F/*. )-F)6%\"\"\"F,F-\"\"#F/-F)6%\"\"$F,F-\"\"\"-F)6%\"\"%F,F-\"\"\"-F)6%\" \"&F,F-\"\"\"-F)6%F.F,F-\"\"\"-%%sqrtG6#*&-F)6%\"\"(F,F-F/F(!\"\"F/FL " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " jac2series(jp,500);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#*&*$),6*$)%\"qG\"$k$\"\"\"\"\"\"*$)F) \"$5#F+!\"\"*$)F)\"#)*F+F,*$)F)\"#GF+F0F,F,*$)F)\"#9F+F0*$)F)\"#qF+F,* $)F)\"$o\"F+F0*$)F)\"$3$F+F,*$)F)\"$!\\F+F0\"\"'F+F+*.),H*$)F)\"$@'F+F 0*$)F)\"$'\\F+F,*$)F)\"$&QF+F0*$)F)\"$)GF+F,*$)F)\"$0#F+F0*$)F)\"$O\"F +F,*$)F)\"#\")F+F0*$)F)\"#SF+F,*$)F)\"#8F+F0F,F,F)F0*$)F)\"#;F+F,*$)F) \"#XF+F0*$)F)\"#))F+F,*$)F)\"$X\"F+F0*$)F)\"$;#F+F,*$)F)\"$,$F+F0*$)F) \"$+%F+F,*$)F)\"$8&F+F0\"\"#F+,H*$)F)\"$.'F+F0*$)F)\"$![F+F,*$)F)\"$r$ F+F0*$)F)\"$w#F+F,*$)F)\"$&>F+F0*$)F)\"$G\"F+F,*$)F)\"#vF+F0*$)F)\"#OF +F,*$)F)\"#6F+F0F,F,*$)F)\"\"$F+F0*$)F)\"#?F+F,*$)F)\"#^F+F0*$)F)\"#'* F+F,*$)F)\"$b\"F+F0*$)F)\"$G#F+F,*$)F)\"$:$F+F0*$)F)\"$;%F+F,*$)F)\"$J &F+F0\"\"\",H*$)F)\"$%fF+F0*$)F)\"$s%F+F,F'F0*$)F)\"$q#F+F,*$)F)\"$!>F +F0*$)F)\"$C\"F+F,*$)F)\"#sF+F0*$)F)\"#MF+F,*$)F)\"#5F+F0F,F,*$)F)\"\" %F+F0*$)F)\"#AF+F,*$)F)\"#aF+F0*$)F)\"$+\"F+F,*$)F)\"$g\"F+F0*$)F)\"$M #F+F,*$)F)\"$A$F+F0*$)F)\"$C%F+F,*$)F)\"$S&F+F0\"\"\",H*$)F)\"$&eF+F0* $)F)\"$k%F+F,*$)F)\"$d$F+F0*$)F)\"$k#F+F,*$)F)\"$&=F+F0*$)F)\"$?\"F+F, *$)F)\"#pF+F0*$)F)\"#KF+F,*$)F)\"\"*F+F0F,F,*$)F)\"\"&F+F0*$)F)\"#CF+F ,*$)F)\"#dF+F0*$)F)\"$/\"F+F,*$)F)\"$l\"F+F0*$)F)\"$S#F+F,*$)F)\"$H$F+ F0*$)F)\"$K%F+F,*$)F)\"$\\&F+F0\"\"\",H*$)F)\"$w&F+F0*$)F)\"$c%F+F,*$) F)\"$]$F+F0*$)F)\"$e#F+F,*$)F)\"$!=F+F0*$)F)\"$;\"F+F,*$)F)\"#mF+F0*$) F)\"#IF+F,*$)F)\"\")F+F0F,F,*$)F)FFF+F0*$)F)\"#EF+F,*$)F)\"#gF+F0*$)F) \"$3\"F+F,*$)F)\"$q\"F+F0*$)F)\"$Y#F+F,*$)F)\"$O$F+F0*$)F)\"$S%F+F,*$) F)\"$e&F+F0\"\"\"-%%sqrtG6#*&,6*$)F)\"$n&F+!\"#*$)F)\"$[%F+\"\"#*$)F) \"$V$F+F\\_l*$)F)\"$_#F+F`_l*$)F)\"$v\"F+F\\_l*$)F)\"$7\"F+F`_l*$)F)\" #jF+F\\_lF4F`_l*$)F)\"\"(F+F\\_lF,F,F+F&!\"\"F+Fc`l" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "It seems that the q-series" } }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&*&&-%!G6$,$%\"qG! \"\"F-6#%\"nG\"\"\")F-,$*&F0F1,&F0F1F1F1F1#F1\"\"#F1\"\"\"&-F*6$F-F-6# ,&F0F7F1F1!\"\"/F0;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "can be written as Jacobi-type product. Assuming that this is t he case we used t.jac2series. to write" }}{PARA 0 "" 0 "" {TEXT -1 55 " this q-series in terms of theta-series at least up to " }{XPPEDIT 18 0 "q^500;" "6#*$%\"qG\"$+&" }{TEXT -1 42 ". This should provide an efficient method" }}{PARA 0 "" 0 "" {TEXT -1 99 "for computing the q- series expansion and also for computing the function at particular val ues of q." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 315 81 "_________________________________________ ________________________________________" }}{PARA 0 "" 0 "" {TEXT 314 12 "EXERCISE 5. " }{TEXT -1 7 " Use " }{TEXT 0 11 "jacprodmake" } {TEXT -1 7 " and " }{TEXT 0 10 "jac2series" }{TEXT -1 38 " to compu te the q-series expansion of" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Sum G6$*&*&&-%!G6$,$%\"qG!\"\"F-6#%\"nG\"\"\")F-,$*&F0F1,&F0\"\"$F1F1F1#F1 \"\"#F1\"\"\"&-F*6$F-F-6#,&F0F8F1F1!\"\"/F0;\"\"!%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 8 "up to " }{XPPEDIT 18 0 "q^1000;" "6#*$%\"qG\"% +5" }{TEXT -1 79 ", assuming it is Jacobi-type product. Can you iden tify the infinite product? " }}{PARA 0 "" 0 "" {TEXT -1 60 "This funct ion occurs in Slater's list [36, Eq.(46), p.156]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 316 81 "_________________________________________ ________________________________________" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 27 "4. The Search for Relations" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "The functions for finding relations between q-series are" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 7 "findhom" }{TEXT -1 3 ", " }{TEXT 0 12 "findhomcombo" } {TEXT -1 2 ", " }{TEXT 0 10 "findnonhom" }{TEXT -1 3 ", " }{TEXT 0 15 "findnonhomcombo" }{TEXT -1 6 ", and " }{TEXT 0 8 "findpoly" } {TEXT -1 1 "." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 4 "4.1 " }{TEXT 0 7 "findhom" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "If the q-series one \+ is concerned with are modular forms of a particular weight, then theor etically these " }}{PARA 0 "" 0 "" {TEXT -1 105 "functions will satisf y homogeneous polynomial relations. See [18, p. 263], for more details and examples." }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{TEXT 0 23 "findhom(L,q,n,topshift)" }{TEXT -1 52 "returns a set of potentia l homogeneous relations of " }}{PARA 0 "" 0 "" {TEXT -1 6 "order " } {TEXT 317 1 "n" }{TEXT -1 32 " among the q-series in the list " } {TEXT 318 1 "L" }{TEXT -1 17 ". The value of " }{TEXT 0 8 "topshift " }{TEXT -1 31 " is usually taken to be zero. " }}{PARA 0 "" 0 "" {TEXT -1 90 "However if it appears that spurious relations are being g enerated then a higher value of " }{TEXT 0 8 "topshift" }}{PARA 0 "" 0 "" {TEXT -1 17 " should be taken." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "The idea is to convert this into a linear algebra problem. Th is program generates a list of monomials of " }}{PARA 0 "" 0 "" {TEXT -1 7 "degree " }{TEXT 319 1 "n" }{TEXT -1 48 " of the functions in the given list of q-series " }{TEXT 320 1 "L" }{TEXT -1 51 ". The q-exp ansion (up to a certain point) of each" }}{PARA 0 "" 0 "" {TEXT -1 104 " monomial is found and converted into a row vector of a matrix. \+ The set of relations is then found by " }}{PARA 0 "" 0 "" {TEXT -1 98 "computing the kernel of the transpose of this matrix. As an exam ple, we now consider relations " }}{PARA 0 "" 0 "" {TEXT -1 29 "betwe en the theta functions " }{XPPEDIT 18 0 "theta[3](q);" "6#-&%&thetaG6 #\"\"$6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "theta[4](q);" "6#-&%& thetaG6#\"\"%6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "theta[3](q^2); " "6#-&%&thetaG6#\"\"$6#*$%\"qG\"\"#" }{TEXT -1 8 ", and " } {XPPEDIT 18 0 "theta[4](q^2);" "6#-&%&thetaG6#\"\"%6#*$%\"qG\"\"#" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(qseri es):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "findhom([theta3(q,1 00),theta4(q,100),theta3(q^2,100),theta4(q^2,100)],q,1,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,#~of~terms~G\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>-----RELATIONS----of~order---G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#<\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "findhom([theta3(q,100),theta4(q,100),theta3(q^2,100), theta4(q^2,1 00)],q,2,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,#~of~terms~G\"#J" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%>-----RELATIONS----of~order---G\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,&*&&%\"XG6#\"\"\"F)&F'6#\"\"#F) F)*$)&F'6#\"\"%F,\"\"\"!\"\",(*$)F&F,F2F)*$)F*F,F2F)*$)&F'6#\"\"$F,F2! \"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "From the session above we \+ see that there is no linear relation between the functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "theta[3](q);" "6#-&%&thetaG6#\" \"$6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "theta[4](q);" "6#-&%&the taG6#\"\"%6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "theta[3](q^2);" " 6#-&%&thetaG6#\"\"$6#*$%\"qG\"\"#" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "theta[4](q^2);" "6#-&%&thetaG6#\"\"%6#*$%\"qG\"\"#" }{TEXT -1 62 ". However, it appears that there are two quadratic relations:" }} }{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%&thetaG6#\"\"$6#*$)%\"qG\" \"#\"\"\"*$-%%sqrtG6#*&,&*$)-F%6#F,F-F.\"\"\"*$)-&F&6#\"\"%F8F-F.F9F.% \"2G!\"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%&thetaG6#\"\"%6#*$)%\"qG\"\"#\"\" \"*$-%%sqrtG6#*&-&F&6#\"\"$6#F,\"\"\"-F%F8F9F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "This is Gauss' parametrization of the arithmetic-ge ometric mean iteration. See [13, Ch 2] for details." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 81 "______________ ___________________________________________________________________" } }{PARA 0 "" 0 "" {TEXT 321 11 "EXERCISE 6." }}{PARA 0 "" 0 "" {TEXT -1 33 "Define a(q), b(q) and c(q) as in " }{TEXT 322 10 "Exercise 2" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 51 "Find homogeneous relatio ns between the functions " }{XPPEDIT 18 0 "a(q);" "6#-%\"aG6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b(q);" "6#-%\"bG6#%\"qG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c(q);" "6#-%\"cG6#%\"qG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a(q^3);" "6#-%\"aG6#*$%\"qG\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b(q^3);" "6#-%\"bG6#*$%\"qG\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c(q^3);" "6#-%\"cG6#*$%\"qG\"\"$" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 27 "In particular, try to get " }{XPPEDIT 18 0 "a(q^3);" "6#-%\"aG6#*$%\"qG\"\"$" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "b(q^3);" "6#-%\"bG6#*$%\"qG\"\"$" }{TEXT -1 14 " in te rms of " }{XPPEDIT 18 0 "a(q);" "6#-%\"aG6#%\"qG" }{TEXT -1 6 " and \+ " }{XPPEDIT 18 0 "b(q);" "6#-%\"bG6#%\"qG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 105 "See [12] for more details. These results lead to a cubic analog of the AGM due to Jon and Peter Borwein " }}{PARA 0 " " 0 "" {TEXT -1 11 "[10], [11]." }}{PARA 0 "" 0 "" {TEXT 324 81 "_____ ______________________________________________________________________ ______" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 4 "4.2 " }{TEXT 0 12 "findh omcombo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{TEXT 0 13 "findhomcombo " }{TEXT -1 16 "is a variant of " }{TEXT 0 7 "findh om" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Suppose" }{TEXT 325 2 " f" }{TEXT -1 19 " is a q-series and " }{TEXT 326 1 "L" }{TEXT -1 23 " is a list of q-series." }}{PARA 0 "" 0 "" {TEXT 0 40 "findhomcomb o(f,L,q,n,topshift,etaoption)" }}{PARA 0 "" 0 "" {TEXT -1 16 "tries to express" }{TEXT 327 3 " f " }{TEXT -1 46 "as a homogeneous polynomial in the members of " }{TEXT 328 1 "L" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 4 "If " }{TEXT 0 14 "etaoption=yes " }{TEXT -1 82 "then ea ch monomial in the combination is ``converted'' into an eta-product u sing " }{TEXT 0 7 "etamake" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We illustrate thi s function with certain Eisenstein series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{TEXT 329 1 "p" }{TEXT -1 21 " an odd prime define \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "chi(m) = ``('m'/'p');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$chiG6#%\"mG-%!G6#*&F'\"\"\"%\"pG !\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "(the Legendre symbol)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Suppose " }{TEXT 330 1 "k" } {TEXT -1 17 " is an integer, " }{XPPEDIT 18 0 "2 <= k;" "6#1\"\"#%\"k G" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "(p-1)/2 = k;" "6#/*&,&%\"pG\" \"\"\"\"\"!\"\"F'\"\"#F)%\"kG" }{TEXT -1 9 " (mod 2)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Define the Eisenstein series" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"UG6$%\"pG%\"kG-%$SumG6$-F*6$*(-%$chiG6#%\"mG\"\"\")%\"nG,&F(F3! \"\"F3F3)%\"qG*&F2F3F5F3F3/F5;F3%)infinityG/F2F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Then \+ " }{XPPEDIT 18 0 "U[p,k];" "6#&%\"UG6$%\"pG%\"kG" }{TEXT -1 30 " is \+ a modular form of weight " }{TEXT 331 1 "k" }{TEXT -1 15 " and charact er " }{XPPEDIT 18 0 "chi;" "6#%$chiG" }{TEXT -1 4 " for" }}{PARA 0 "" 0 "" {TEXT -1 24 "the congruence subgroup " }{XPPEDIT 18 0 "Gamma[0](p );" "6#-&%&GammaG6#\"\"!6#%\"pG" }{TEXT -1 73 ". See [28], [20] for mo re details. The classical result is the following " }}{PARA 0 "" 0 "" {TEXT -1 54 "identity found by Ramanujan [32, Eq. (1.52), p. 354]:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"UG6$\"\"&\"\"#*&*$)-%$etaG6#,$%$tauGF'F'\"\"\"F1-F-6#F0!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Kolberg [28] has found many re lations between such Eisenstein series and certain eta products." }} {PARA 0 "" 0 "" {TEXT -1 17 "The eta function " }{XPPEDIT 18 0 "eta(ta u);" "6#-%$etaG6#%$tauG" }{TEXT -1 29 " is a modular form of weight " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 37 " [27, p.121]. Hence the modular forms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"\"*&*$)-%$etaG6#,$ %$tauG\"\"$\"\"&\"\"\"F2-F,6#F/!\"\"" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"#*&*$)-%$etaG6#%$tauG\"\"&\"\"\"F0-F,6#,$F .F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "are modular forms of weight " }{XPPEDIT 18 0 "(5-1)/2 = 2;" "6#/*&,&\"\"&\"\"\"\"\"\"!\"\"F'\"\"#F)\"\"#" } {TEXT -1 55 ". In fact, it can be shown that they are modular forms " }}{PARA 0 "" 0 "" {TEXT -1 3 "on " }{XPPEDIT 18 0 "Gamma[0](5);" "6#-& %&GammaG6#\"\"!6#\"\"&" }{TEXT -1 16 " with character " }{XPPEDIT 18 0 "``(`.`/`5`);" "6#-%!G6#*&%\".G\"\"\"%\"5G!\"\"" }{TEXT -1 36 " . W e might therefore expect that " }{XPPEDIT 18 0 "U[5,6];" "6#&%\"UG6$ \"\"&\"\"'" }{TEXT -1 16 " can be written" }}{PARA 0 "" 0 "" {TEXT -1 37 "as a homogeneous cubic polynomial in " }{XPPEDIT 18 0 "B[1];" " 6#&%\"BG6#\"\"\"" }{TEXT -1 55 ",B[2]. We write a short maple progra m to compute the " }}{PARA 0 "" 0 "" {TEXT -1 19 "Eisenstein series \+ " }{XPPEDIT 18 0 "U[p,k];" "6#&%\"UG6$%\"pG%\"kG" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 " with(numtheory): " }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " UE:=proc(q,k,p,trunk) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " local x,m,n: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " x:=0: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " for m f rom 1 to trunk do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " for n from 1 to trunk/m do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " \+ x:=x + L(m,p)*n^(k-1)*q^(m*n): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " od: \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " end: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{TEXT 0 15 "UE(q,k,p,trunk)" }{TEXT -1 27 "returns the q-expansion of " }{XPPEDIT 18 0 "U[p,k];" "6#&%\"UG 6$%\"pG%\"kG" }{TEXT -1 12 " up through " }{XPPEDIT 18 0 "q^trunk;" "6 #)%\"qG%&trunkG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We not e that " }{TEXT 0 6 "L(m,p)" }{TEXT -1 29 "returns the Legendre symbol " }{XPPEDIT 18 0 "``(m/p);" "6#-%!G6#*&%\"mG\"\"\"%\"pG!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "We are now ready to stud y " }{XPPEDIT 18 0 "U[5,6];" "6#&%\"UG6$\"\"&\"\"'" }{TEXT -1 1 "." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " with(qseries): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " f := UE(q,6,5,50): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " B1 := etaq(q,1,50)^5/etaq (q,5,50): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " B2 := q*et aq(q,5,50)^5/etaq(q,1,50): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " findhomcombo(f,[B1,B2],q,3,0,yes);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,#~of~terms~G\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% R-----possible~linear~combinations~of~degree------G\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<#,(*&)-%$etaG6#,$%$tauG\"\"&\"\"$\"\"\")-F(6#F+ \"\"*F.\"\"\"*&)F'F2F.)F0F-F.\"#S*&*$)F'\"#:F.F.*$)F0\"\"$F.!\"\"\"$N$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,(*&)&%\"XG6#\"\"\"\"\"#\"\"\"&F (6#F+F*F**&F'F*)F-F+F,\"#S*$)F-\"\"$F,\"$N$" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 58 "It would appear that \+ " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"UG6$\"\"&\"\"',(*& )-%$etaG6#,$%$tauGF'\"\"$\"\"\")-F-6#F0\"\"*F2\"\"\"*&)F,F6F2)F4F1F2\" #S*&*$)F,\"#:F2F2*$)F4\"\"$F2!\"\"\"$N$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The proof is a straightforward exercise using the theory \+ of modular forms." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 333 81 "_________________________________________________ ________________________________" }}{PARA 0 "" 0 "" {TEXT 332 11 "EXER CISE 7." }}{PARA 0 "" 0 "" {TEXT -1 34 "Define the following eta produ cts:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"\"*&*$)-%$etaG6#, $%$tauG\"\"(F0\"\"\"F1-F,6#F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"CG6#\"\"#*&)-%$etaG6#%$tauG\"\"$\"\"\")-F+6#,$F-\"\"(F.F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6#\"\"$*&*$)-%$etaG6#%$tauG\"\" (\"\"\"F0-F,6#,$F.F/!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 43 "What is the \+ weight of these modular forms? " }}{PARA 0 "" 0 "" {TEXT -1 6 "Write \+ " }{XPPEDIT 18 0 "U[7,3];" "6#&%\"UG6$\"\"(\"\"$" }{TEXT -1 13 " in te rms of " }{XPPEDIT 18 0 "C[1];" "6#&%\"CG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "C[2];" "6#&%\"CG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "C[3];" "6#&%\"CG6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "The identity that you should find was originally due to Ramanuj an." }}{PARA 0 "" 0 "" {TEXT -1 47 "Also see Fine [15, p. 159] and [19 , Eq. (5.4)]." }}{PARA 0 "" 0 "" {TEXT -1 26 "If you are ambitious fin d " }{XPPEDIT 18 0 "U[7,9];" "6#&%\"UG6$\"\"(\"\"*" }{TEXT -1 15 " in \+ terms of " }{XPPEDIT 18 0 "C[1];" "6#&%\"CG6#\"\"\"" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "C[2];" "6#&%\"CG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "C[3];" "6#&%\"CG6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 334 81 "________________________________________________________ _________________________" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 4 "4.3 " }{TEXT 0 10 "findnonhom" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "In section 4.1 we introduced the f unction " }{TEXT 0 8 "findhom " }{TEXT -1 47 "to find homogeneous rela tions between q-series." }}{PARA 0 "" 0 "" {TEXT -1 29 "The nonhomogen eous analog is " }{TEXT 0 10 "findnonhom" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The syntax of " }{TEXT 0 11 "findnonhom \+ " }{TEXT -1 16 "is the same as " }{TEXT 0 7 "findhom" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Typically (but not necessarily) " } {TEXT 0 8 "findhom " }{TEXT -1 61 "is used to find relations between m odular forms of a certain " }}{PARA 0 "" 0 "" {TEXT -1 66 "weight. To \+ find relations between modular functions we would use " }{TEXT 0 10 " findnonhom" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "We consider an example involving theta functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(qseries): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F := q -> theta3(q,500)/theta3(q^5,100): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "U := 2*q*theta(q^10,q^25,5)/theta3( q^25,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG,$*&*&%\"qG\"\"\",8 *$)F(\"$v&\"\"\"F)*$)F(\"$g$F.F)*$)F(\"$&>F.F)*$)F(\"#!)F.F)*$)F(\"#:F .F)F)F)*$)F(\"#NF.F)*$)F(\"$?\"F.F)*$)F(\"$b#F.F)*$)F(\"$S%F.F)*$)F(\" $v'F.F)F)F.,.*$)F(\"$D'F.\"\"#*$)F(\"$+%F.FN*$)F(\"$D#F.FN*$)F(\"$+\"F .FN*$)F(\"#DF.FNF)F)!\"\"FN" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "EQNS := findnonhom([F(q),F(q^5),U],q,3,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,#~of~terms~G\"#h" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% >-----RELATIONS----of~order---G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%EQNSG<#,,\"\"\"F'&%\"XG6#\"\"$!\"\"*(&F)6#F'F'&F)6#\"\"#F'F(F'F'* $)F0F2\"\"\"F,*$)F(F2F5F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ANS:=EQNS[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ANSG,,\"\"\"F& &%\"XG6#\"\"$!\"\"*(&F(6#F&F&&F(6#\"\"#F&F'F&F&*$)F/F1\"\"\"F+*$)F'F1F 4F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CHECK := subs(\{X[1] =F(q),X[2]=F(q^5),X[3]=U\},ANS): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(CHECK,q,500);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"qG-%\"OG6#\"\"\"\"$+&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "W e define" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"FG6#%\"qG*&-& %&thetaG6#\"\"$F&\"\"\"-F*6#*$)F'\"\"&F.!\"\"" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"UG6#%\"qG,$*&-%$SumG6$)F',(*$)%\"nG\"\"#\"\" \"\"#DF1\"#5\"\"\"F6/F1;,$%)infinityG!\"\"F:F3-&%&thetaG6#\"\"$6#*$)F' F4F3!\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "We note that U(q) \+ and F(q) are modular functions since they are ratios of theta series. " }}{PARA 0 "" 0 "" {TEXT -1 51 "From the session above we see that it appears that " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"\"F%* (-%\"FG6#%\"qGF%-F(6#*$)F*\"\"&\"\"\"F%-%\"UGF)F%F%,(*$)F+\"\"#F0F%*$) F1F6F0F%F1F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Observe how we we re able to verify this equation to high order." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 5 "When " }{TEXT 0 11 "findnonhom " }{TEXT -1 61 "retu rns a set of relations the variable X has been declared " }{TEXT 335 6 "global" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 79 "This is so \+ we can manipulate the relations. It this way we were able to assign " }{TEXT 0 3 "ANS" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "to the \+ relation found and then use " }{TEXT 0 5 "subs " }{TEXT -1 4 "and " } {TEXT 0 6 "series" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "to ch eck it to order " }{XPPEDIT 18 0 "O(q^500);" "6#-%\"OG6#*$%\"qG\"$+&" }{TEXT -1 1 "." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 4 "4.4 " }{TEXT 0 15 "findnonhomcombo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The syntax of " }{TEXT 0 16 "findnonhomcombo " }{TEXT -1 15 "is the same as " } {TEXT 0 12 "findhomcombo" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "We consider an example involving eta functions. First we define \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xiG*&-%$etaG6#,$%$tauG\"#\\\"\"\"-F'6#F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG*&*$)-%$etaG6#,$%$tauG\"\"(\"\"%\"\"\"F/*$)-F)6#F,\"\"%F/! \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Using the theory of modul ar functions it can be shown that one must be able to write " } {XPPEDIT 18 0 "T^2;" "6#*$%\"TG\"\"#" }{TEXT -1 14 " in terms of T" }} {PARA 0 "" 0 "" {TEXT -1 31 "and powers of xi. We now use " }{TEXT 0 15 "findnonhomcombo" }{TEXT -1 8 " to get " }{XPPEDIT 18 0 "T^2;" "6 #*$%\"TG\"\"#" }{TEXT -1 14 " in terms of " }{XPPEDIT 18 0 "xi;" "6#% #xiG" }{TEXT -1 8 " and T." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(qseries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "xi:=q ^2*etaq(q,49,100)/etaq(q,1,100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "T:=q*(etaq(q,7,100)/etaq(q,1,100))^4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "findnonhomcombo(T^2,[T,xi],q,7,-10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,#~of~terms~G\"#Z" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%+matrix~is~G\"#P%$~x~G\"#Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%R-----possible~linear~combinations~of~degree------G\" \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$*$)&%\"XG6#\"\"\"\"\"#\"\"\", 6*&F&F))&F'6#F*\"\"$F+\"#\\*$)F/\"\"'F+\"$V$*$)F/\"\"&F+\"$Z\"F/F)*$F. F+\"#@*$)F/F*F+\"\"(*&F&F+F>F+\"#N*&F&F+F/F)F?*$)F/F?F+F6*$)F/\"\"%F+F 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Then it seems that" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"TG\"\"#\"\"\",2*&,(%#xiG\"\"(*$)F,F'F (\"#N*$)F,\"\"$F(\"#\\\"\"\"F&F5F5*$)F,F-F(\"$V$F,F5F1\"#@*$)F,\"\"%F( F4*$)F,\"\"&F(\"$Z\"*$)F,\"\"'F(F8F.F-" }}{PARA 0 "" 0 "" {TEXT -1 95 "This is the modular equation used by Watson [41] to prove Ramanujan's partition congruences for" }}{PARA 0 "" 0 "" {TEXT -1 79 "powers of 7 . Also see [5] and [26]., and see [16] for an elementary treatment." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 342 88 "____________________________________ ____________________________________________________" }}{PARA 0 "" 0 " " {TEXT 336 11 "EXERCISE 8." }}{PARA 0 "" 0 "" {TEXT -1 6 "Define" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xiG*&-%$etaG6#,$%$tauG\"#D\"\"\"-F' 6#F*!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG*&*$)-%$etaG6#,$%$tauG\"\"&\"\"'\"\"\"F/*$)-F)6#F ,\"\"'F/!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 0 16 "findno nhomcombo " }{TEXT -1 32 "to express T as a polynomial in " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 39 " of degree 5. The modular equation \+ you " }}{PARA 0 "" 0 "" {TEXT -1 99 "find was used by Watson to prove \+ Ramanujan's partition congruences for powers of 5. See [23]for an " }} {PARA 0 "" 0 "" {TEXT -1 20 "elementarytreatment." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 340 88 "_________________ ______________________________________________________________________ _" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 337 11 "EXERCISE 9." }}{PARA 0 "" 0 "" {TEXT -1 28 "Define a(q) and c(q) as in " }{TEXT 338 10 "Exercis e 2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Define " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"xG6#%\"qG*&*$)-%\"cGF&\"\"$\"\"\"F.*$)- %\"aGF&\"\"$F.!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 52 "and the classical \+ Eisenstein series (usually called " }{XPPEDIT 18 0 "E[3];" "6#&%\"EG6# \"\"$" }{TEXT -1 19 "; see [35, p. 93])" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"NG6#%\"qG,&\"\"\"F)-%$SumG6$*&*&)%\"nG\"\"&\"\"\")F'F0F)F2, &F)F)F3!\"\"!\"\"/F0;F)%)infinityG!$/&" }}{PARA 0 "" 0 "" {TEXT -1 5 " Use " }{TEXT 0 16 "findnonhomcombo " }{TEXT -1 42 "to express N(q) in terms of a(q) and x(q)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 339 4 "HINT" }{TEXT -1 87 ": N(q) is a modular form of weight 6 and a(q) \+ and c(q) are modular forms of weight 1. " }}{PARA 0 "" 0 "" {TEXT -1 42 "See [8] for this result and many others." }}{PARA 0 "" 0 "" {TEXT 341 88 "________________________________________________________ ________________________________" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 4 "4.5 " }{TEXT 0 8 "findpoly" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 " The function " }{TEXT 0 8 "findpoly" }{TEXT -1 79 " is used to find a \+ polynomial relation between two given q-series with degrees " }}{PARA 0 "" 0 "" {TEXT -1 10 "specified." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 31 "findpoly(x,y,q,deg1,deg2,check)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 " returns a possible polynomial in X, Y (with corresponding de grees deg1, deg2) which is satisfied by " }}{PARA 0 "" 0 "" {TEXT -1 13 "the q-series " }{TEXT 343 1 "x" }{TEXT -1 5 " and " }{TEXT 344 1 " y" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 0 5 "check" }{TEXT -1 45 " is assigned then the relation is checked to " }{XPPEDIT 18 0 "O(q^check);" "6#-%\"OG6#)%\"qG%&checkG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "We illustrate this funct ion with an example involving theta functions and the function a(q) an d c(q) " }}{PARA 0 "" 0 "" {TEXT -1 15 "encountered in " }{TEXT 345 11 "Exercises 2" }{TEXT -1 6 " and " }{TEXT 346 1 "7" }{TEXT -1 22 ". It can be shown that" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" aG6#%\"qG,&*&-&%&thetaG6#\"\"$F&\"\"\"-F+6#*$)F'F.\"\"\"F/F/*&-&F,6#\" \"#F&F/-F7F1F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "See [12] for details. This equation provides a better way of computing the q-serie s expansion of a(q) " }}{PARA 0 "" 0 "" {TEXT -1 27 "than the definiti on. In " }{TEXT 347 10 "Exercise 2" }{TEXT -1 26 " you would have f ound that" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"cG6#%\"qG,$* &*$)-%$etaG6#,$%$tauG\"\"$F1\"\"\"F2-F-6#F0!\"\"F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "See [12]. for a proof. Define " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG*&*$)%\"cG\"\"$\"\"\"F**$)%\"aG \"\"$F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "and " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,&*&*$)-&%&thetaG6#\"\"#6#%\"qGF -\"\"\"F0*$)-F*6#*$)F/\"\"$F0\"\"#F0!\"\"\"\"\"*&*$)-&F+6#F7F.F-F0F0*$ )-F?F4\"\"#F0F9F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "We use " } {TEXT 0 9 "findpoly " }{TEXT -1 38 "to find a polynomial relation betw een " }{TEXT 348 1 "x" }{TEXT -1 5 " and " }{TEXT 349 1 "y" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(qseries):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "x1 := radsimp(theta2(q,100)^ 2/theta2(q^3,40)^2): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "x2 := theta3(q,100)^2/theta3(q^3,40)^2: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x := x1+x2: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "c := q*etaq(q,3,100)^9/etaq(q,1,100)^3: " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 72 "a := radsimp(theta3(q,100)*theta3(q^3,40)+thet a2(q,100)*theta2(q^3,40)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "c := 3*q^(1/3)*etaq(q,3,100)^3/etaq(q,1,100): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "y := radsimp(c^3/a^3): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P1:=findpoly(x,y,q,3,1,60);" }}{PARA 6 "" 1 "" {TEXT -1 24 "WARNING: X,Y are global." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%'~dims~G\"\")\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% 2The~polynomial~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&),&%\"XG\" \"\"\"\"'F(\"\"$\"\"\"%\"YGF(F(*$),&F'F(\"\"#F(F0F+!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2Checking~to~orderG\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"qG-%\"OG6#\"\"\"\"#f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,&*&),&%\"XG\"\"\"\"\"'F*\"\"$\"\"\"%\"YGF*F**$),&F)F*\"\" #F*F2F-!#F" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "It seems that " } {TEXT 350 1 "x" }{TEXT -1 5 " and " }{TEXT 351 1 "y" }{TEXT -1 21 " sa tisfy the equation" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&**-% \"pG6$%\"xG%\"yG\"\"\"%\"=GF+),&F)F+\"\"'F+\"\"$\"\"\"F*F+F+*$),&F)F+ \"\"#F+F5F1!#F\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Therefore \+ it would seem that " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&*$)%\"cG\"\" $\"\"\"F)*$)%\"aG\"\"$F)!\"\",$*&*$),&%\"xG\"\"\"\"\"#F5F6F)F)*$),&F4F 5\"\"'F5\"\"$F)F.\"#F" }}{PARA 0 "" 0 "" {TEXT -1 42 "See [8, pp. 4237 --4240] for more details." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 354 88 "________________________________________ ________________________________________________" }}{PARA 0 "" 0 "" {TEXT 352 12 "EXERCISE 10." }{TEXT -1 10 " Define " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG*&*$)-&%&thetaG6#\"\"$6#%\"qG\"\"#\"\"\"F0*$)- F)6#*$)F.F,F0\"\"#F0!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 0 9 "polyfind " }{TEXT -1 8 "to find " }{XPPEDIT 18 0 "y = c^3/(a^3); " "6#/%\"yG*&%\"cG\"\"$*$%\"aG\"\"$!\"\"" }{TEXT -1 28 " as a rationa l function in " }{TEXT 353 1 "m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The answer is Eq.(12.8) in [8]." }}{PARA 0 "" 0 "" {TEXT 355 88 "______________________________________________________________ __________________________" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 3 "" 0 "" {TEXT -1 23 "5. Sifting coefficients" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Suppose we are given a q-series " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"qG-%$SumG6$*&&% \"aG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 53 "Occasionally it will turn out the generating function" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"aG6#,&*&%\"mG \"\"\"%\"nGF-F-%\"rGF-F-)%\"qGF.F-/F.;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "will have a very nice form. A famous exam ple for p(n) is due to Ramanujan:" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&-%\"pG6#,&%\"nG\"\"&\"\"%\"\"\"F/)%\"qGF,F/ /F,;\"\"!%)infinityG,$-%(ProductG6$*&*$),&F/F/)F1,$F,F-!\"\"\"\"'\"\" \"FB*$),&F/F/F0F@\"\"&FB!\"\"/F,;F/F5F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "See [1, Cor. 10.6]. In fact, G.H. Hardy and Major MacMah on [31, p. xxxv] both agreed that this is " }}{PARA 0 "" 0 "" {TEXT -1 36 "Ramanujan's most beautiful identity." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 8 "Suppose " }{TEXT 356 1 "s" }{TEXT -1 16 " is the q-serie s" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$SumG6$*&&%\"aG6#%\"i G\"\"\")%\"qGF+F,/F+;\"\"!%)infinityGF,-%\"OG6#)F.%\"TGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "then " }{TEXT 0 15 "sift(s,q,n,k,T)" } {TEXT -1 21 " returns the q-series" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$SumG6$*&&%\"aG6#,&*&%\"nG\"\"\"%\"iGF.F.%\"kGF.F.) %\"qGF/F./F/;\"\"!%)infinityGF.-%\"OG6#)F2*&%\"TG\"\"\"F-!\"\"F." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "We illustrate this function with \+ another example from the theory of partitions. Let pd(n) denote the " }}{PARA 0 "" 0 "" {TEXT -1 73 "number of partitions of n into distinct parts. Then it is well known that" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%$SumG6$*&-%#pdG6#%\"nG\"\"\")%\"qGF,F-/F,;\"\"!%)i nfinityGF-%\"=GF--%(ProductG6$,&F-F-F.F-/F,;F-F3F--F66$*&,&F-F-)F/,$F, \"\"#!\"\"\"\"\",&F-F-F.FB!\"\"F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "We now examine the generating function of pd(5n+1) in maple." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(qseries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PD:=series(etaq(q,2,200)/etaq(q,1,2 00),q,200): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "PD1:=sift(P D,q,5,1,199);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$PD1G,\\p\"\"\"F&% \"qG\"\"%*$)F'\"#I\"\"\"\")?@z?*$)F'\"#9F,\"&#*H$*$)F'F(F,\"#w*$)F'\" \"#F,\"#7*$)F'\"#5F,\"%(4%*$)F'\"#CF,\"(+)oB*$)F'\"#EF,\"()o5]*$)F'\"# :F,\"&]K&*$)F'\"\"(F,\"$o'*$)F'\"\"&F,\"$l\"*$)F'F8F,\"&w?\"*$)F'\"#AF ,\"(Wx3\"*$)F'\"#NF,\"*\\0U4\"*$)F'\"\"$F,\"#K*$)F'\"\"'F,\"$S$*$)F'\" \")F,\"%g7*$)F'\"\"*F,\"%/B*$)F'\"#6F,\"%3r*$)F'\"#8F,\"&K,#*$)F'\"#;F ,\"&cZ)*$)F'\"#F,\"')y<$*$)F'\"#? F,\"'IL[*$)F'\"#@F,\"'g#G(*$)F'\"#BF,\"()Q6;*$)F'\"#DF,\"(FqX$*$)F'\"# FF,\"(Wc@(*$)F'\"#GF,\")crK5*$)F'\"#HF,\")WUp9*$)F'\"#JF,\")g\\EH*$)F' FinF,\")SD)4%*$)F'\"#LF,\")W[6d*$)F'\"#MF,\")w'H#z*$)F'\"#OF,\"*oNZ]\" *$)F'\"#PF,\"*'4%31#*$)F'\"#QF,\"*[!Q6G*$)F'\"#RF,\"*oe2#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "etamake(PD1,q,38);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&)-%$etaG6#,$%$tauG\"\"&\"\"$\"\"\")-F'6#,$F*\" \"#F2F-F-*()%\"qG#\"\"&\"#CF--F'6#,$F*\"#5\"\"\")-F'6#F*\"\"%F-!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "So it would seem that" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&-%#pdG6#,&%\"nG\"\"&\"\"\"F.F.)% \"qGF,F./F,;\"\"!%)infinityG-%(ProductG6$*&*&),&F.F.)F0,$F,F-!\"\"\"\" $\"\"\"),&F.F.)F0,$F,\"\"#F>FEF@F@*&,&F.F.)F0,$F,\"#5F>\"\"\"),&F.F.F/ F>\"\"%F@!\"\"/F,;F.F4" }}{PARA 0 "" 0 "" {TEXT -1 50 "This result was found originally by Rodseth [33]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 358 88 "______________________________ __________________________________________________________" }}{PARA 0 "" 0 "" {TEXT 357 12 "EXERCISE 11." }{TEXT -1 88 " Rodseth also foun d the generating functions for pd(5n+r) for r=0, 1, 2, 3 and 4. For " }}{PARA 0 "" 0 "" {TEXT -1 12 "each r use " }{TEXT 0 5 "sift " } {TEXT -1 5 "and " }{TEXT 0 12 "jacprodmake " }{TEXT -1 60 "to identif y these generating functions as infinite products." }}{PARA 0 "" 0 "" {TEXT 359 88 "________________________________________________________ ________________________________" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 22 "6. Product Identities" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "At present, the package contains the Triple Product identity, t he Quintuple Product identity and " }}{PARA 0 "" 0 "" {TEXT -1 82 "Win quist's identity. These are the most commonly used of the Macdonald i dentities" }}{PARA 0 "" 0 "" {TEXT -1 97 "[30], [37], [38]. The Macd onald identities are the analogs of the Weyl denominator for affine \+ " }}{PARA 0 "" 0 "" {TEXT -1 102 "roots systems. Hopefully, a later v ersion of this package will include these more general identities." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 32 "6.1 The Triple Product Identity" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The triple product identity is" }}{PARA 0 "" 0 "" {TEXT -1 5 "(6.7)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sumG6$*()!\"\"%\"mG \"\"\")%\"zGF*F+)%\"qG,$*&F*F+,&F*F+F)F+F+#F+\"\"#F+/F*;,$%)infinityGF )F8-%(ProductG6$*(,&F+F+*&F-F+)F/,&%\"nGF+F)F+F+F)F+,&F+F+*&)F/FA\"\" \"F-!\"\"F)F+,&F+F+FDF)F+/FA;F+F8" }}{PARA 0 "" 0 "" {TEXT -1 8 "where " }{XPPEDIT 18 0 "z <> 0;" "6#0%\"zG\"\"!" }{TEXT -1 73 " and |q| <1. The Triple Product Identity is originally due to Jacobi " }} {PARA 0 "" 0 "" {TEXT -1 94 "[24,Vol I]. The first combinatorial proo f of the triple product identity is due to Sylvester " }}{PARA 0 "" 0 "" {TEXT -1 85 "[39]. Recently, Andrews [3] and Lewis [29] have fo und nice combinatorial proofs. " }}{PARA 0 "" 0 "" {TEXT -1 96 "The tr iple product occurs frequently in the theory of partitions. For insta nce, most proofs of " }}{PARA 0 "" 0 "" {TEXT -1 79 "the Rogers-Ramanu jan identity crucially depend on the triple product identity. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 18 "tripleprod(z,q,T) " }{TEXT -1 44 " returns the q-series expansi on to order " }{XPPEDIT 18 0 "O(q^T);" "6#-%\"OG6#)%\"qG%\"TG" } {TEXT -1 5 " of " }}{PARA 0 "" 0 "" {TEXT -1 95 "Jacobi's triple prod uct (6.7). This expansion is found by simply truncating the right si de of" }}{PARA 0 "" 0 "" {TEXT -1 6 "(6.7)." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "triplepro d(z,q,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,<*&*$)%\"qG\"#@\"\"\"F) *$)%\"zG\"\"'F)!\"\"\"\"\"*&*$)F'\"#:F)F)*$)F,\"\"&F)F.!\"\"*&*$)F'\"# 5F)F)*$)F,\"\"%F)F.F/*&*$)F'\"\"'F)F)*$)F,\"\"$F)F.F7*&*$)F'\"\"$F)F)* $)F,\"\"#F)F.F/*&F'F)F,F.F7F/F/F,F7*&)F,\"\"#F)F'F/F/*&)F,FIF)FHF)F7*& )F,\"\"%F)FAF)F/*&)F,\"\"&F)F:F)F7*&)F,FBF)F2F)F/" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "tripleprod(q,q^3,10); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,<*$)%\"qG\"#d\"\"\"\"\"\"*$)F&\"#SF(!\"\"*$)F&\"#EF( F)*$)F&\"#:F(F-*$)F&\"\"(F(F)*$)F&\"\"#F(F-F)F)F&F-*$)F&\"\"&F(F)*$)F& \"#7F(F-*$)F&\"#AF(F)*$)F&\"#NF(F-*$)F&\"#^F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The last calculation is an illustration of Euler's P entagonal Number Theorem " }}{PARA 0 "" 0 "" {TEXT -1 19 "[1, Cor. 1.7 p.11]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "(6.8)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(-%(ProductG6$,&\"\"\"F))%\"qG%\"nG!\"\"/F,;F)%)in finityGF)%\"=GF)-F&6$*(,&F)F))F+,&F,\"\"$F-F)F-F),&F)F))F+,&F,F8!\"#F) F-F),&F)F))F+,$F,F8F-F)F.F)-%$SumG6$*&)F-F,F))F+,$*&F,F)F7F)#F)\"\"#F) /F,;,$F0F-F0" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 35 "6.2 The Quintupl e Product Identity" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The followi ng identity is the Quintuple Product Identity:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "(6.9)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,&-%\"~G6$ ,$%\"zG!\"\"%\"qG6#%)infinityG\"\"\"&-F'6$,$*&F,\"\"\"F*!\"\"F+F,F-F/& -F'6$*&)F*\"\"#F5F,F/*$)F,F 0;" "6#0%\"zG\"\"!" } {TEXT -1 24 ". This identity is the " }{XPPEDIT 18 0 "BC[1];" "6#&%#B CG6#\"\"\"" }{TEXT -1 42 " case of the Macdonald identities [30]. " }}{PARA 0 "" 0 "" {TEXT -1 85 "The quintuple product identity is usual ly attributed to Watson [40]. However it can" }}{PARA 0 "" 0 "" {TEXT -1 84 "be found in Ramanujan's lost notebook [32, p. 207]. Als o see [7] for more history" }}{PARA 0 "" 0 "" {TEXT -1 16 "and some p roofs." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The function " }{TEXT 0 16 "quinprod(z,q,T) " }{TEXT -1 58 "returns the quintuple product id entity in different forms:" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 100 " \+ (i) If T is a positive integer it returns the q-expansion of the r ight side of (6.9) to order " }{XPPEDIT 18 0 "O(q^T);" "6#-%\"OG6#)% \"qG%\"TG" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 16 " (ii) If T= " } {TEXT 360 6 "prodid" }{TEXT -1 8 " then " }{TEXT 0 20 "quinprod(z,q, prodid)" }{TEXT -1 56 " returns the quintuple product identity in prod uct form." }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 18 " (iii) If T = \+ " }{TEXT 361 8 "seriesid" }{TEXT -1 8 " then " }{TEXT 0 22 "quinprod (z,q,seriesid)" }{TEXT -1 56 " returns the quintuple product identity in series form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(qs eries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "quinprod(z,q,pro did);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*,&-%\"~G6$,$%\"zG!\"\"%\"qG 6#%)infinityG\"\"\"&-F'6$,$*&F,\"\"\"F*!\"\"F+F,F-F/&-F'6$*&)F*\"\"#F5 F,F/*$)F,FF5F5*$)F*\"\"$F5F6*$)F,\"\"$F5F-F/&-F'6$*&F,F5)F*FUF5FSF-F/&- F'6$FSFSF-F/F/**F*F/&-F'6$*&F,F5*$)F*\"\"$F5F6FSF-F/&-F'6$*&F>F5FZF5FS F-F/FenF5F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "quinprod(z,q ,seriesid);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,&-%\"~G6$,$%\"zG!\" \"%\"qG6#%)infinityG\"\"\"&-F'6$,$*&F,\"\"\"F*!\"\"F+F,F-F/&-F'6$*&)F* \"\"#F5F,F/*$)F,F " 0 "" {MPLTEXT 1 0 16 "qu inprod(z,q,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*&,&*$)%\"zG\"#7\" \"\"\"\"\"*&F*F**$)F(\"#6F*!\"\"F+F+)%\"qG\"#AF*F+*&,&*$)F(\"\"*F*!\" \"*&F*F**$)F(\"\")F*F0F9F+)F2F)F*F+*&,&*$)F(\"\"'F*F+*&F*F**$)F(\"\"&F *F0F+F+)F2\"\"&F*F+*&,&*$)F(\"\"$F*F9*&F*F**$)F(\"\"#F*F0F9F+F2F+F+F+F +F(F+*&,&*&F*F**$)F(\"\"$F*F0F9*$)F(\"\"%F*F9F+)F2\"\"#F*F+*&,&*&F*F** $)F(\"\"'F*F0F+*$)F(\"\"(F*F+F+)F2F`oF*F+*&,&*&F*F**$)F(\"\"*F*F0F9*$) F(\"#5F*F9F+)F2\"#:F*F+*&,&*&F*F**$)F(\"#7F*F0F+*$)F(\"#8F*F+F+)F2\"#E F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Let's examine a more int eresting application. Euler's infinite product may be dissected accor ding to the" }}{PARA 0 "" 0 "" {TEXT -1 36 " residue of the exponent o f q mod 5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Product(1-q^n ,n=1..infinity)=E[0](q^5)+q*E[1](q^5)+q^2*E[2](q^5)+q^3*E[3](q^5)+q^4* E[4](q^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(ProductG6$,&\"\"\"F( )%\"qG%\"nG!\"\"/F+;F(%)infinityG,,-&%\"EG6#\"\"!6#*$)F*\"\"&\"\"\"F(* &F*F(-&F36#F(F6F(F(*&)F*\"\"#F:-&F36#FAF6F(F(*&)F*\"\"$F:-&F36#FGF6F(F (*&)F*\"\"%F:-&F36#FMF6F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "By (6.8) we see that " }{XPPEDIT 18 0 "E[3] = E[4];" "6#/&%\"EG6#\"\"$ &F%6#\"\"%" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 11 " si nce " }{XPPEDIT 18 0 "n*(3*n-1)/2 = 0;" "6#/*&*&%\"nG\"\"\",&*&\"\"$ F'F&F'F'\"\"\"!\"\"F'F'\"\"#F,\"\"!" }{TEXT -1 44 ", 1 or 2 mod 5. Let 's see if we can identify" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "E[0];" "6#&%\"EG6#\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(qseries): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "EULER:=etaq(q,1,500): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "E0:=sift(EULER,q,5,0,499);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E0G,@\"\"\"F&%\"qGF&*$)F'\"\"$\"\"\"!\"\"*$)F'\"\"(F +F,*$)F'\"\")F+F,*$)F'\"#9F+F,*$)F'\"#?F+F&*$)F'\"#HF+F&*$)F'\"#JF+F&* $)F'\"#UF+F&*$)F'\"#_F+F,*$)F'\"#mF+F,*$)F'\"#pF+F,*$)F'\"#&)F+F,*$)F' \"#**F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "jp:=jacprodmak e(E0,q,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#jpG*&*&-%$JACG6%\"\" #\"\"&%)infinityG\"\"\"-F(6%\"\"!F+F,F-\"\"\"-F(6%F-F+F,!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "jac2prod(jp);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&*(&-%\"~G6$*$)%\"qG\"\"&\"\"\"F)6#%)infinityG \"\"\"&-F'6$*$)F+\"\"#F-F)F.F0&-F'6$*$)F+\"\"$F-F)F.F0F-*&&-F'6$F+F)F. \"\"\"&-F'6$*$)F+\"\"%F-F)F.\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "qp:=quinprod(q,q^5,20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(qp,q,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# +A%\"qG\"\"\"\"\"!F%\"\"\"!\"\"\"\"$F(\"\"(F(\"\")F(\"#9F%\"#?F%\"#HF% \"#JF%\"#UF(\"#_F(\"#mF(\"#pF(\"#&)-%\"OG6#F%\"#**" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "From our maple session it appears that" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "(6.10)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"EG6#\"\"!*&*(&-%\"~G6$*$)%\"qG\"\"&\"\"\"F.6#%)inf inityG\"\"\"&-F,6$*$)F0\"\"#F2F.F3F5&-F,6$*$)F0\"\"$F2F.F3F5F2*&&-F,6$ F0F.F3\"\"\"&-F,6$*$)F0\"\"%F2F.F3\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "and that this product can be gotten by replacing " } {TEXT 362 1 "q" }{TEXT -1 5 " by " }{XPPEDIT 18 0 "q^5;" "6#*$)%\"qG \"\"&\"\"\"" }{TEXT -1 6 " and " }{TEXT 363 1 "z" }{TEXT -1 4 " by " }{TEXT 364 1 "q" }{TEXT -1 38 " in the product side of the quintuple \+ " }}{PARA 0 "" 0 "" {TEXT -1 24 "product identity (6.9)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 366 88 "________ ______________________________________________________________________ __________" }}{PARA 0 "" 0 "" {TEXT 365 12 "EXERCISE 12." }{TEXT -1 89 " (i) Use the quintuple product identity (6.9) and Euler's pen tagonal number theorem " }}{PARA 0 "" 0 "" {TEXT -1 16 "to prove (6.10 )." }}{PARA 0 "" 0 "" {TEXT -1 64 "(ii) Use maple to identify and pr ove product expressions for " }{XPPEDIT 18 0 "E[1];" "6#&%\"EG6#\"\" \"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "E[2];" "6#&%\"EG6#\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 79 "(iii) This time see if you can repeat (i), (ii) but split the exponent mod 7." }}{PARA 0 "" 0 "" {TEXT -1 97 "(iv) Can you generalize these results to arbitrary modulus? Atkin and Swinnerton-Dyer found a " }}{PARA 0 "" 0 "" {TEXT -1 35 "generalization. See Lemma 6 in [6]." }}{PARA 0 "" 0 "" {TEXT 367 88 "________________________________________________________ ________________________________" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 24 "6.3 Winquist's Identity" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Back in 1969, Lasse Winquist [4 3] discovered a remarkable identity" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "(6.11)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*4&-%!G6$%\"aG%\"qG6# %)infinityG\"\"\"&-F'6$*&F*\"\"\"F)!\"\"F*F+F-&-F'6$%\"bGF*F+F-&-F'6$* &F*F2F7F3F*F+F-&-F'6$*&F)F-F7F-F*F+F-&-F'6$*&F*F2*&F)\"\"\"F7\"\"\"F3F *F+F-&-F'6$*&F)F2F7F3F*F+F-&-F'6$*&*&F*F-F7F2F2F)F3F*F+F-)&-F'6$F*F*F+ \"\"#F2-%$SumG6$-FV6$*()!\"\",&%\"nGF-%\"mGF-F-,&*&,&)F),$Fhn!\"$F-)F) ,&Fhn\"\"$FboF-FfnF-,&)F7,$FinF_oF-)F7,&FinFboF-F-FfnF-F-*&,&)F),&FinF _oF-F-F-)F),&FinFboFTF-FfnF-,&)F7,&FhnFboFTF-F-)F7,&FhnF_oFfnF-FfnF-F- F-)F*,&*&FhnF-,&FhnF-F-F-F-#FboFT*&FinF-FgoF-#F-FTF-/Fin;,$F,FfnF,/Fhn ;\"\"!F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "By dividing both side s by " }{XPPEDIT 18 0 "(1-a)*(1-b);" "6#*&,&\"\"\"\"\"\"%\"aG!\"\"F&, &\"\"\"F&%\"bGF(F&" }{TEXT -1 16 " and letting " }{TEXT 368 9 "a, b --> " }{TEXT -1 36 "1 he was able to express the product" }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Product(1-q^n,n = 1 .. infinity) ^10;" "6#*$)-%(ProductG6$,&\"\"\"\"\"\")%\"qG%\"nG!\"\"/F-;\"\"\"%)inf inityG\"#5F*" }}{PARA 0 "" 0 "" {TEXT -1 62 " as a double series and p rove Ramanujan's partition congruence" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "p(11*n+6) = 0;" "6#/-%\"pG6#,&*&\"#6 \"\"\"%\"nGF*F*\"\"'F*\"\"!" }{TEXT -1 12 " (mod 11)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 " This was probably the first truly elementary proof of Ramanujan's cong ruence modulo 11. The interested " }}{PARA 0 "" 0 "" {TEXT -1 108 "rea der should see Dyson's article [14] for some fascinating history on id entities for powers of the Dedekind" }}{PARA 0 "" 0 "" {TEXT -1 105 " \+ eta function and how they led to the Macdonald identities. A new proo f of Winquist's identity has been " }}{PARA 0 "" 0 "" {TEXT -1 102 "fo und recently by S.-Y. Kang [25]. Mike Hirschhorn [22] has found a fou r-parameter generalization of " }}{PARA 0 "" 0 "" {TEXT -1 20 "Winquis t's identity." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The function " }{TEXT 0 18 "winquist(a,b ,q,T) " }{TEXT -1 52 "returns the q-expansion of the right side of (6 .11)" }}{PARA 0 "" 0 "" {TEXT -1 9 "to order " }{XPPEDIT 18 0 "O(q^T); " "6#-%\"OG6#)%\"qG%\"TG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We close with an \+ example. For " }{XPPEDIT 18 0 "1 <= k;" "6#1\"\"\"%\"kG" }{TEXT -1 12 " < 33 define" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"QG6#% \"kG-%(ProductG6$*&,&\"\"\"F-)%\"qG,(%\"nG\"$j$F'\"#6!$j$F-!\"\"F-,&F- F--)F/,&F1F2F'!#66#,&F-F-)F/,$F1F2F5F5F-/F';F-%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "N ow define the following functions:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "A[0] = Q(15);" "6#/&%\"AG6#\"\"!-%\"QG6#\"#:" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "A[3] = Q(12);" "6#/&%\"AG6#\"\"$-%\"QG6# \"#7" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "A[7] = Q(6);" "6#/&%\"AG6# \"\"(-%\"QG6#\"\"'" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "A[8] = Q(3); " "6#/&%\"AG6#\"\")-%\"QG6#\"\"$" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "A[9] = Q(9);" "6#/&%\"AG6#\"\"*-%\"QG6#\"\"*" }{TEXT -1 1 ";" }}} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"!,&-%\"QG6#\"#;\" \"\"*&)%\"qG\"\"#\"\"\"-F*6#\"\"&F-!\"\"" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"\",&-%\"QG6#\"#9F'*&%\"qGF'-F*6#\"\")F'!\" \"" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"#,&-%\"QG6# \"#8\"\"\"*&)%\"qG\"\"$\"\"\"-F*F&F-!\"\"" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"%,&-%\"QG6#\"\"(\"\"\"*&%\"qGF--F*F&F-F-" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"\"(,&-%\"QG6#\"#5 \"\"\"*&)%\"qG\"\"$\"\"\"-F*6#F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "These functions occur in Theorem 6.7 of [17] as well as the fun ction A_0B_2 - q^2 A_9B_4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(qseries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Q:=n- >tripleprod(q^n,q^33,10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A0:=Q(15): A3:=Q(12): A7:=Q(6):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "A8:=Q(3): A9:=Q(9): " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "B2:=Q(13)-q^3*Q(2): B4:=Q(7)+q*Q(4): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "IDG:=series( ( A0*B2-q^2*A9*B4),q,2 00): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(IDG,q,10); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"qG\"\"\"\"\"!!\"\"\"\"#!\"#\" \"$F%\"\"&F%\"\"(F%\"\"*-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "jp:=jacprodmake(IDG,q,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#jpG*&*(-%$JACG6%\"\"#\"#6%)infinityG\"\"\")-F(6%\"\" $F+F,F*\"\"\"-F(6%\"\"&F+F,F-F2*$)-F(6%\"\"!F+F,\"\"#F2!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "jac2prod(jp);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*0&-%\"~G6$*$)%\"qG\"\"#\"\"\"*$)F*\"#6F,6#%)inf inityG\"\"\"&-F&6$*$)F*\"\"*F,F-F0F2)&-F&6$F-F-F0F+F,)&-F&6$*$)F*\"\"$ F,F-F0F+F,)&-F&6$*$)F*\"\")F,F-F0F+F,&-F&6$*$)F*\"\"&F,F-F0F2&-F&6$*$) F*\"\"'F,F-F0F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "series(w inquist(q^5,q^3,q^11,20),q,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+=% \"qG\"\"\"\"\"!!\"\"\"\"#!\"#\"\"$F%\"\"&F%\"\"(F%\"\"*F%\"#6F%\"#7F' \"#8F'\"#:F'\"#;F'\"#=-%\"OG6#F%\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "series(IDG-winquist(q^5,q^3,q^11,20),q,60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"qG-%\"OG6#\"\"\"\"#\\" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "From our maple session it seems that" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "(6.12)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&&%\"AG6#\"\"!\"\"\"&%\"BG6#\"\"#F*F**()%\"qGF.\"\" \"&F'6#\"\"*F*&F,6#\"\"%F*!\"\"*0&-%\"~G6$*$F0F2*$)F1\"#6F26#%)infinit yGF*&-F=6$*$)F1F5F2F@FCF*)&-F=6$F@F@FCF.F2)&-F=6$*$)F1\"\"$F2F@FCF.F2) &-F=6$*$)F1\"\")F2F@FCF.F2&-F=6$*$)F1\"\"&F2F@FCF*&-F=6$*$)F1\"\"'F2F@ FCF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "and that this product app ears in Winquist's identity on replacing" }}{PARA 0 "" 0 "" {TEXT -1 5 "q by " }{XPPEDIT 18 0 "q^11;" "6#*$)%\"qG\"#6\"\"\"" }{TEXT -1 15 " and letting " }{XPPEDIT 18 0 "a = q^5;" "6#/%\"aG*$)%\"qG\"\"&\"\" \"" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "b = q^3;" "6#/%\"bG*$)%\"q G\"\"$\"\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 88 "______________________________________ __________________________________________________" }}{PARA 0 "" 0 "" {TEXT 369 11 "EXERCISE 13" }{TEXT -1 103 ". (i) Prove (6.12) by usi ng the triple product identity (6.7) to write the right side of Winqui st's " }}{PARA 0 "" 0 "" {TEXT -1 48 "identity eqn.winquist. as a sum \+ of two products." }}{PARA 0 "" 0 "" {TEXT -1 60 "(ii) In a similar m anner find and prove a product form for" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%\"AG6#\"\"!\"\"\"&%\"BGF'F)F)*()%\"qG\"\"$\"\"\"&F&6#\"\"( F)&F+6#\"\"%F)!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 371 88 "__ ______________________________________________________________________ ________________" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 10 "REFERENCES" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "[1]George E. Andrews. The theory of partitions. Addison-W esley Publishing Co.," }}{PARA 0 "" 0 "" {TEXT -1 73 " Reading, Mass .-London-Amsterdam, 1976. Encyclopedia of Mathematics and" }}{PARA 0 " " 0 "" {TEXT -1 28 " its Applications, Vol. 2." }}{PARA 0 "" 0 "" {TEXT -1 76 "[2]George E. Andrews. 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Combi-" }}{PARA 0 "" 0 "" {TEXT -1 34 " natorial Theory, 6:56-59, 1969." }}}}{MARK "0 1 0" 12 } {VIEWOPTS 1 1 0 3 2 1804 }