DATE: Tuesday, April 20 (2010), at 4:05pm
PLACE: LIT 305
SPEAKER:
Krishnaswami Alladi
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TITLE:
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Vanishing coefficients in the expansion of products of Rogers-Ramanujan type
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ABSTRACT:
By a Rogers-Ramanujan type product, we mean an expression
F(M,r,s;q)=Π m in S(1-qm) /
Π m in T(1-qm)
where S and T are the set of positive integers congruent to ± r
and ± s mod M respectively. The name Rogers-Ramanujan type product
is because F(5,2,1;q) is the celebrated product associated with
Ramanujan's continued fraction. Let us write the product as an infinite
power series in q with coefficients cn. In 1978 Richmond and Szekeres
proved that the c 4n+3=0 for the product F(8,3,1;q). This product is
the one associated with the Göllnitz-Gordon continued fractiion. They used
the circle method of Hardy-Ramanujan to estimate the coefficients cn.
In 1979 Andrews and Bressoud generalized the result of Richmond and
Szekeres by showing that in the expansion of the product F(2k,r,k-r;q),
the coefficients cn equal to 0 when n lies in a certain arithmetic
progression mod k. They did not use the circle method but instead used the
spectacular
1Ψ1
summation of Ramanujan.
In 1994 Gordon and I improved
the Andrews-Bressoud modulus 2k result in the case k even by showing that
two sequences of coefficients in certain linear combinations of the products F
1/F become zero, one of which is the sequence identified by Andrews-Bressoud.
Our improved result applies to the modulus 8 product treated by Richmond
and Szekeres. We also use the
1Ψ1
summation of Ramanujan but in addition
we exploit transformation properties of certain quadratic forms. Our approach also
yields a general result for Rogers-Ramanujan products with modulus jk - see
K. Alladi and B. Gordon, "Vanishing coefficients in the expansion of products
of Rogers-Ramanujan type", in Proc. Rademacher Centenary Conference,
(G. E. Andrews and D. Bressoud, Eds.), Contemp. Math. 166, (1994), 129-139.
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