On arithmetic nature of a $q$-Euler-double zeta values

Abstract:

Chatterjee and Garg [Proc. Amer. Math. Soc. 151 (2023), pp. 2011-2022] established closed form for a $q$-analogue of the Euler-Stieltjes constants. In this article, we aim to build upon their work by extending it to a $q$-analogue of the double zeta function. Specifically, we derive a closed form expression for $\gamma _{0,0}(q)$ which is a $q$-analogue of Euler’s constant of height $2$ and appear as the constant term in the Laurent series expansion of a $q$-analogue of the double zeta function around $s_1 = 1$ and $s_2=1$.

Moreover, we examine the linear independence of a set of numbers involving the constant $\gamma _0^{\prime *}(q^i)$, where $1 \leq i \leq r$ for any integer $r \geq 1$, that appears in the Laurent series expansion of a $q$-double zeta function. Finally, we discuss the irrationality of certain numbers involving a $2$-double Euler-Stieltjes constant ($\gamma _{0,0}(2)$).