Number Theory Seminar

DATE: Tuesday, October 4 (2011), at 1:55pm  
PLACE: LIT 368
 
SPEAKER: Alexander Berkovich
 
TITLE: Sign Reversing Involutions. Three pedagogical examples.
 

ABSTRACT:
This is a pedagogical introduction to sign reversing involution technique. I will illustrate the power of this technique by establishing three formulas below.

  1. q-binomial identity
    $$ \sum_{k=0}^n (-1)^k q^{k(k-n)+n(n+1)/2} \left[\begin{matrix} n \\ k \end{matrix}\right]_q^2 = (-1)^{n/2} q^{ n(n+2)/4 } \left[\begin{matrix} n \\ n/2 \end{matrix}\right]_{q^2} $$ if n is even, and $$ \sum_{k=0}^n (-1)^k q^{k(k-n)+n(n+1)/2} \left[\begin{matrix} n \\ k \end{matrix}\right]_q^2 =0, $$ if n is odd.
  2. Euler's Pentagonal Number Theorem:
    $$ \prod_{n=1}^\infty (1 - q^n) = \sum_{k=-\infty}^\infty (-1)^k q^{k(3k+1)/2}. $$
  3. Andrews-Garvan (1988) identity relating vector partitions with prescribed cranks with ordinary partitions with prescribed cranks.


 

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Last update made Mon Oct 3 21:16:28 EDT 2011.