On a \(q\)-deformation of modular forms (Q2633811)

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.