Hecke modular forms and q-Hermite polynomials
The author uses a technique of L. J. Rogers, expansion in terms of q- Hermite polynomials \(A_ n(\cos \theta | q)=\sum^{n}_{i=0}\left[ \begin{matrix} n\\ i\end{matrix} \right]\) cos(n-2i)\(\theta\), where \(\left[\begin{matrix} n\\ i\end{matrix} \right]=\prod^{i}_{j=1}(i-q^{n-i+j})/(1-q^ j)\) is the Gaussian polynomial, to derive a number of identities which express a summation of the form \[ \sum_{(n,m)\in D}(-1)^{f(n,m)} q^{Q(n,m)+L(n,m)} \] as a rational product of \(\eta\)-functions, where Q is a quadratic form, L is linear and \(D\subseteq \{(n,m)\in {\mathbb{Z}}\times {\mathbb{Z}}|\) Q(n,m)\(\geq 0\}\). The most famous identities of this type are Jacobi's identity and Hecke's identity expressing \[ \prod_{n\geq 1}(1-q^ n)^ 3=\sum_{n\geq 0}(-1)^ n(2n+1)q^{(n^ 2+n)/2}=\sum^{\infty}\quad_{m=-\infty}\sum_{n\geq | m|}(- 1)^ nq^{(n^ 2+n)/2} \] and, respectively, \[ \prod_{n\geq 1}(1- q^ n)^ 2=\sum^{\infty}_{m=-\infty}\sum^{\infty}_{n\geq 2| m|}(-1)^{n\quad +m} q^{(n^ 2-3m^ 2+n+m)/2}. \] A typical identity is the following (one of the three identities) \[ (q;q)^ 2_{\infty} \sum_{m\geq 0}q^ mf(m)= \] \[ \sum^{\infty}_{m=- \infty}\sum_{n\geq 2| m|}(-1)^{n+m} q^{(n^ 2-3m^ 2+n+m)/2}\sum^{| m|}_{k=0}(q^{-m};q)_ k (q^ m;q)_ k q^ kf(k) \] where f(m) is an arbitrary sequence and \[ (a;q)_{\infty}=\prod (1-aq^ i),\quad i\geq 0,\quad (a;q)_ x=(a,q)_{\infty}/(aq^ x;q)_{\infty},\quad x\in {\mathbb{R}}. \] Other corollaries of these identities are derived using Heine's summation. Similar results have been studied by various authors using different techniques including Lie algebras.
scientific article; zbMATH DE number 3922659