On ${\varvec{q}}$-congruences related to ${\varvec{H}}_{\varvec{n}}\left( {\varvec{x;q}}\right)$ and ${\varvec{M}}_{{\varvec{n}}}\left( {\varvec{x;q}}\right)$

Abstract

In this paper, we examine $\sum \limits _{k=m}^{p-1}q^{k} {k \brack m} _{q}H_{k}\left( x;q\right) \pmod {\left[ p\right] _{q}}$ and $M_{p-1}\left( x;q\right) \pmod {\left[ p\right] _{q}},$ where for real number x,  $H_{n}\left( x;q\right) =\sum \limits _{k=1}^{n}\frac{x^{k}}{\left[ k\right] _{q}}$ and $M_{n}\left( x;q\right) =\sum \limits _{k=1}^{n}k\frac{x^{k}}{\left[ k\right] _{q}}.$ For example, for an odd prime number p and $m\in \left\{ 0,1,...,p-2\right\} ,$

$$\begin{aligned}{} & {} \left[ m+1\right] _{q}\sum \limits _{k=m}^{p-1}q^{k} {k \brack m} _{q}H_{k}\left( x;q\right) \\{} & {} \quad \equiv \left( -1\right) ^{m}q^{-\left( {\begin{array}{c}m\\ 2\end{array}}\right) }\left( 1-\left( x;q\right) _{p}\right) -\frac{q^{m}}{\left( xq;q\right) _{m}}x^{m+1}\left[ p\right] _{x} \pmod {\left[ p\right] _{q}}. \end{aligned}$$