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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]

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]

We consider two sequences a(n) and b(n), 1\leq n<\infty, generated by Dirichlet series of the forms \begin{equation*} \sum _{n=1}^{\infty }\dfrac {a(n)}{\lambda _n^{s}}\qquad \text {and}\qquad \sum _{n=1}^{\infty }\dfrac {b(n)}{\mu _n^{s}}, \end{equation*} satisfying a familiar functional equation involving the gamma function \Gamma (s). A general identity is established. Appearing on one side is an infinite series involving a(n) and modified Bessel functions K_{\nu }, wherein on the other side is an infinite series involving b(n) that is an analogue of the Hurwitz zeta function. Six special cases, including a(n)=\tau (n) and a(n)=r_k(n), are examined, where \tau (n) is Ramanujan's arithmetical function and r_k(n) denotes the number of representations of n as a sum of k squares. All but one of the examples appear to be new. [ABSTRACT FROM AUTHOR]

There are a number of papers in the literature which contain Cesàro analogues of results already known for martingale sums of Vilenkin-Fourier series. We show that for Vilenkin systems of bounded type, these are not merely analogues but actually generalizations. Indeed, we prove that convergence of the Cesàro means of a Vilenkin series $S$ implies convergence of martingale partial sums of $S$ itself. [ABSTRACT FROM AUTHOR]

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence $x$ satisfies $\lim Cx = L$ and $\Delta x_k = O (1/k)$, then $\lim x = L$. In this paper it is shown that the hypothesis $\lim Cx = L$ can be replaced by the weaker assumption of the statistical limit: st-lim $Cx = L$, i.e., for every $\epsilon >0$, $\lim n^{-1} | { k \leq n: | ( Cx)_k - L | \geq \epsilon } | = 0$. Similarly, the ``one-sided'' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing $\lim Cx $ and $\lim Ax$ with st-lim $Cx$ and st-lim $Ax$, respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing $\lim_t (Bx)_t=L$, where $t$ is a continuous parameter, with $\lim_n (Bx)_n =L$, and further replacing it by $(B^{*})$-st-lim $B^{*} x =L$, where $B^{*}$ is the Borel matrix method. [ABSTRACT FROM AUTHOR]

Any coded subshift XC defined by a set C of code words contains a subshift, which we call LC, consisting of limits of single code words. We show that when C satisfies the unique decipherability property, the topological entropy h(XC) of XC is determined completely by h(LC) and the number of code words of each length. More specifically, we show that h(XC) = h(LC) exactly when a certain infinite series is less than or equal to 1, and when that series is greater than 1, we give a formula for h(XC). In the latter case, an immediate corollary (using a result from [Israel J. Math. 192 (2012), pp. 785-817] is that XC has a unique measure of maximal entropy. [ABSTRACT FROM AUTHOR]

In a recent article published in 2017, Barroso, P'erez, and Popescu-Pampu employ the Lagrange inversion formula to solve certain Newton-Puiseux equations when the solutions to the inverse problems are given. More precisely, for an irreducible f(x, y) ∊ K[[x, y]] over an algebraically closed field K of characteristic zero, they calculate the coefficients of η(x1/n) which would meet f(x, η(x1/n)) = 0 in terms of the coefficients of ξ(y1/m) that satisfy f(ξ(y1/m), y) = 0. This article will present an alternative approach to solving the problem using diagonalizations on polynomial sequences of binomial-type. Along the way, a close relationship between binomial-type sequences and the Lagrange inversion formula will be observed. In addition, it will extend the result to give the coefficients of (x1/n) directly in terms of the coefficients of f(x, y). As an application, an infinite series formula for the roots of complex polynomials will be obtained together with a sufficient condition for its convergence. [ABSTRACT FROM AUTHOR]

We present the strongest known knot invariant that can be computed effectively (in polynomial time). [ABSTRACT FROM AUTHOR]

For a dynamical system (X, f), X being a compact metric space with metric d and f being a continuous map X →X, a set S ⊆ X is scrambled if every pair (x, y) of distinct points in S is scrambled, i.e., ... The system (X, f) is Li-Yorke chaotic if it has an uncountable scrambled set. It is known that for interval and circle maps, the existence of a scrambled pair implies Li-Yorke chaos, in fact, the existence of a Cantor scrambled set. We prove that the same result holds for graph maps. We further show that on compact countable metric spaces one scrambled pair implies the existence of an infinite scrambled set. [ABSTRACT FROM AUTHOR]

We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type ƒ(I). H are equidistributed (as p tends to infinity), where ƒ is a polynomial, I is an interval of Fp and H is an approximate subgroup of F*p with size larger than polylogarithmic in p. [ABSTRACT FROM AUTHOR]