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On a \(q\)-deformation of modular forms. (English) Zbl 1445.11014

The purpose of this article is to give examples of \(q\)-deformations of hypergeometric evaluations, which are linked with the coefficients of modular forms rather than Dirichlet characters.
Furthermore, several almost \(q\)-hypergeometric congruences modulo (powers of) cyclotomic polynomials in \(q\) are proved or conjectured. It starts with the slowly-converging Ramanujan-type identity
\[\sum_{k=0}^{\infty}\frac{(\frac 12)_k^3}{k!^3}=\frac{\pi}{\Gamma(\frac 34)^4}.\tag{*} \]
A \(q\)-extension of (*) which accommodates the related congruences \(\bmod~p^2\) is given.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
33C05 Classical hypergeometric functions, \({}_2F_1\)
11F03 Modular and automorphic functions
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References:

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