Congruences between binomial coefficients \(\biggl({{2f}\atop f} \biggr)\) and Fourier coefficients of certain \(\eta{}\)-products (Q1210056)

By extending a method of the reviewer [J. Number Theory 25, 201-210 (1987; Zbl 0614.10011)], the author proves the following result. Let \(k, \ell\in \mathbb{N}\) with \(\text{gcd} (k,\ell)=1\). Let \(p\) be a prime of the form \(p= kf+\ell\). Consider the power series \[ \sum_{n\geq 1} \gamma_ n^{(k,\ell)} q^ n= \eta(k\tau)^ 2 \eta(2k \tau)^{m+1} \eta(4k \tau)^{3- 3m} \eta (8k\tau )^{2m-2} \] where \(\eta(\tau)\) is the Dedekind \(\eta\)-function with \(q= e^{2 \pi i\tau}\). Then we have \[ \left( \begin{smallmatrix} 2f\\ f\end{smallmatrix} \right) \equiv (-1)^ f \gamma_ p^{(k,\ell)} \pmod p. \]