Determinant Expressions for $q$-Harmonic Congruences and Degenerate Bernoulli Numbers

Abstract

The generalized harmonic numbers $H_n^{(k)}=\sum_{j=1}^n j^{-k}$ satisfy the well-known congruence $H_{p-1}^{(k)}\equiv 0\pmod{p}$ for all primes $p\geq 3$ and integers $k\geq 1$. We derive $q$-analogs of this congruence for two different $q$-analogs of the sum $H_n^{(k)}$. The results can be written in terms of certain determinants of binomial coefficients which have interesting properties in their own right. Furthermore, it is shown that one of the classes of determinants is closely related to degenerate Bernoulli numbers, and new properties of these numbers are obtained as a consequence.