On 𝑞-analogue of Euler–Stieltjes constants

Chatterjee, Tapas; Garg, Sonam

doi:10.1090/proc/16288

On $q$-analogue of Euler–Stieltjes constants
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by Tapas Chatterjee and Sonam Garg

Proc. Amer. Math. Soc. 151 (2023), 2011-2022

DOI: https://doi.org/10.1090/proc/16288

Published electronically: February 17, 2023

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Abstract:

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a $q$-analogue of the Euler constant and proved the irrationality of certain numbers involving $q$-Euler constant. In this paper, we improve their results and prove the linear independence result involving $q$-analogue of the Euler constant. Further, we derive the closed-form of a $q$-analogue of the $k$-th Stieltjes constant $\gamma _k(q)$. These constants are the coefficients in the Laurent series expansion of a $q$-analogue of the Riemann zeta function around $s=1$. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series $\sum _{n\geq 1}{\sigma _1(n)}/{t^n}$ for any integer $t > 1$. Finally, we study the transcendence nature of some infinite series involving $\gamma _1(2)$.

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