Zeros of generalized Rogers-Ramanujan series: asymptotic and combinatorial properties
The author studies the Rogers-Ramanujan series \[ R(x,q):=\sum_{n=0}^{\infty}\frac{q^{n^2}x^n}{(q;q)_n},\quad (\alpha;q)_n:=\prod_{k=0}^{n-1}(1-\alpha q^k),\quad|q|<1. \] In his famous Lost Notebook, Ramanujan mentioned a Hadamard product expansion of the form \[ R(x,q)=\prod_{n=1}^{\infty}\left(1+\frac{xq^{2n-1}}{1-\sum_{i=1}^{\infty}y_iq^{ni}}\right) \] with explicit formulas for the first four coefficients \(\{y_i\}_{i=1}^4\) in terms of the function \[ \psi(q):=\sum_{n=0}^{\infty}q^{n(n+1)/2}=\frac{(q^2;q^2)_{\infty}}{(q;q^2)_{\infty}}. \] It is shown that for \(q\) tending to \(1\) the coefficients \(y_n\) tend to a rational multiple of \(\pi^{-1}\) for \(n\) odd, and to a rational multiple of \(\pi^{-2}\) for \(n\) even. This is a slightly different form of a conjecture given in [\textit{M. E. H. Ismail} and \textit{C. Zhang}, Adv. Math. 209, No. 1, 363--380 (2007; Zbl 1107.33018)]. It is also shown that the coefficients \(y_i\) are connected to a \(q\)-analogue of the tangent numbers \(T_n\) appearing in the Taylor expansion \[ \tan x=\sum_{n=0}^{\infty}\frac{T_{2n+1}x^{2n+1}}{(2n+1)!}. \] Finally, a connection with Jacobian elliptic functions leads to several recursion formulas for the calculation of series expansions for the zeros of \(R(x,q)\). This is motivated by the earlier mentioned work of Ismail and Zhang.
scientific article; zbMATH DE number 5722074
scientific article