INFINITE series (Mathematics), ARITHMETIC mean, FOURIER series, MATHEMATICS, SEQUENCE spaces, SUMMABILITY theory
Abstract
Quite recently, we have obtained two main theorems dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series [C. R. Math. Acad. Sci. Paris 359 (2021), pp. 323–328]. In this paper, we have generalized these theorems for a general summability method. We have also obtained some new and known results for certain absolute summability methods. [ABSTRACT FROM AUTHOR]
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). [ABSTRACT FROM AUTHOR]