Finite $q$-multiple harmonic sums with one exceptional index

Abstract

In this paper, we give explicit expressions about $q$-harmonic sums on $1-\cdots -1,A,1-\cdots -1$ indices. The case $A=1$ has been extensively studied, and a variety of identities, explicit evaluations, and structural properties are known. Likewise, numerous explicit evaluations are available for $q$-multiple zeta values and $q$-harmonic sums with uniform indices $A-\cdots -A$. Although several methods are available for treating non-uniform indices, explicit evaluations for the pattern $1-\cdots -1,A,1-\cdots -1$ remain comparatively scarce. The case $A=2$ was settled only recently. Here we extend this framework to the general case $A\ge 3$ and obtain explicit formulas in terms of binomial coefficients and degenerate Bernoulli numbers.