## 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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