On 𝑞-analogue of Euler–Stieltjes constants

Abstract

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q q -analogue of the Euler constant and proved the irrationality of certain numbers involving q q -Euler constant. In this paper, we improve their results and prove the linear independence result involving q q -analogue of the Euler constant. Further, we derive the closed-form of a q q -analogue of the k k -th Stieltjes constant γ k ( q ) \gamma _k(q) . These constants are the coefficients in the Laurent series expansion of a q q -analogue of the Riemann zeta function around s = 1 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 ∑ n ≥ 1 σ 1 ( n ) / t n \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1 t > 1 . Finally, we study the transcendence nature of some infinite series involving γ 1 ( 2 ) \gamma _1(2) .