Some congruences for modulus 13 related to partition generating function

Abstract

Let \(p(n)\) be the classical partition function, i.e., the number of non-increasing sequences of positive integers whose sum is \(n\). Let \[ F^{(k,m)} =q^k \sum_{n=0}^\infty p(mn+k) y^{n},\quad (z; q)_\infty =\prod_{n=1}^\infty (1-zq^{n-1} ), \] \[ P(0)=(y^m; y^m)_\infty , \quad P(a)= (y^a; y^m)_\infty (y^{m-a}; y^m)_\infty \text{ if } m\nmid a, \] where \(y=q^m\). In this paper, using Kolberg's method, the authors give congruences for \(F^{(k,13)}\) modulo \(13\) for all \(0\leq k\leq 12\). For example, \[ F^{(0,13)}\equiv y\frac{P(2)}{P(4)}\left\{ \frac{P(0)P(6)}{yP(1)P(3)}-4(y; y)^{11}_{\infty} \right\} \pmod{13} . \]