Square Series Generating Function Transformations

Abstract

We construct new integral representations for transformations of the ordinary generating function for a sequence, $⟨f_n⟩$, into the form of a generating function that enumerates the corresponding "square series" generating function for the sequence, $⟨q^{n^2}{f}_n⟩$, at an initially fixed non-zero $q\in \mathbb{C}$. The new results proved in the article are given by integral-based transformations of ordinary generating function series expanded in terms of the Stirling numbers of the second kind. We then employ known integral representations for the gamma and double factorial functions in the construction of these square series transformation integrals. The results proved in the article lead to new applications and integral representations for special function series, sequence generating functions, and other related applications. A summary Mathematica notebook providing derivations of key results and applications to specific series is provided online as a supplemental reference to readers.

Keywords: Square series, generating function, series transformation, integral transformation, gamma function, double factorial, Stirling number of the second kind, Jacobi theta function, Ramanujan theta function

References in corpus (1)

• Asymptotic expansions for enumerating connected labelled graphs

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