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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]

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]

In this paper we construct an infinite family of paramodular forms of weight 2 which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta block conjecture of Gritsenko-Poor-Yuen (2013) related to the most important infinite series of theta blocks of weight 2 and q-order 1. We also consider some applications of this result. [ABSTRACT FROM AUTHOR]

Let Q = {qk}k=1 be a sequence of positive integers with qk = 2 for every k = 1. Then each point x ∈ [0, 1] is attached with an infinite series expansion ... which is called the Cantor series expansion of x. In this paper, we study the shrinking target problems for the system induced by the Cantor series expansion. More precisely, put TQn (x) = q1 · · · qnx-[q1 · · · qnx]; the shrinking target problem in such a nonautonomous system can be formulated as considering the size of the set ... where y is a fixed point in [0, 1] and ϕ : N → (0, 1) is a positive function with ϕ(n) → 0 as n → ∞. It is proved that both the Lebesgue measure and the Hausdorff measure of Ey(ϕ) fulfill a dichotomy law according to the divergence or convergence of certain series. [ABSTRACT FROM AUTHOR]