Series with Binomial-Like Coefficients for Evaluation and 3D Visualization of Zeta Functions

Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

“…The present research extends the investigations of the asymptotics for Delannoy numbers undertaken by Noble [9,10] and Wang, Zheng and Chen [11] (as well as our research into central and local limit theorems for combinatorial numbers satisfying a class of triangular arrays [12][13][14][15]). Noble has obtained asymptotic expansions for the central weighted Delannoy numbers (u r,r ) and the numbers along the the diagonal with slope 2 (u r,2r ).…”

Central Limit Theorems for Combinatorial Numbers Associated with Laguerre Polynomials

“…Apart from the theoretical value (generating functions are a very important tool to derive the identities, connections, and interpolation functions for polynomials, or limit theorems for corresponding combinatorial numbers), these results can be applied to the construction of efficient algorithms for the calculation of the values of special functions. We have used similar limit theorems for the combinatorial numbers in calculations of the Riemann zeta function (see Theorem 3 in [14] and Algorithm 3 in [17]). Moreover, the presented asymptotic normality results may have also an important utilization in choosing a suitable cumulative distribution function or a cumulative intensity function for models in insurance [18].…”

“…Limit theorems for numbers satisfying a class of triangular arrays can be established using properties of ordinary or semi-exponential generating functions (cf. [13,14]). Let Ω n be an integral random variable with the probability mass function…”

Series with Binomial-Like Coefficients for the Investigation of Fractal Structures Associated with the Riemann Zeta Function

“…This algorithm, introduced in [4], also uses series (3) (case j = 2), but with different binomial-like coefficients, c…”

“…In this paper, we continue the study of efficient algorithms for the computation 8 of the Riemann zeta function over the complex plane, introduced by Borwein [7] 9 and extended by Belovas [1], Belovas and Sabaliauskas [4], Belovas, Sakalauskas and Throughout this paper, we denote by Φ(x) the cumulative distribution function of the standard normal distribution, and by Φ(x) we denote the corresponding tail distribution Φ(x) = 1 − Φ(x). Γ(s), B(x, y) and W(x) denote the gamma function, the beta function and the Lambert W function respectively.…”