Asymptotic formulas for g-gonal sequence factorials

In this note, we provide basic asymptotic formulas for approximating large g-gonal sequence factorials by using Stirling and Burnside asymptotic approximation formulas for large factorials. More accurate asymptotic approximation formulas for large g-gonal sequence factorials resulting from some recent, more accurate asymptotic formulas for large factorials that have appeared in the literature are presented.     

Asymptotic formulas for sequence factorial of arithmetic progression

This note provides asymptotic formulas for approximating the sequence factorial of members of a finite arithmetic progression by using Stirling, Burnside and other more accurate asymptotic formulas for large factorials that have appeared in the literature.     

Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.     

Zero-Eigenvalue Eigenfunctions for Differences of Elliptic Relativistic Calogero-Moser Hamiltonians

Letting A_l(x) denote the commuting analytic difference operators of elliptic relativistic Calogero-Moser type, we present and study zero-eigenvalue eigenfunctions for the operators A_l(x) − A_l(−y) (with l = 1, 2,..., N and x, y ∈ ℂ ^N). The eigenfunctions are products of elliptic gamma functions. They are invariant under permutations of x₁,..., x_N and y₁,..., y_N and under interchange of the step-size parameters. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 31–41, January, 2006.     

Helping students build a path of understanding from ratio and proportion to decimal notation

Various studies have shown that students of all levels struggle to understand decimal numbers. This paper discusses a novel approach to increasing students’ conceptual understanding of decimal numbers. Rather than approach decimal notation as a discrete and separate mathematical topic, this approach enables students to work with contextual problems to gain a solid understanding of ratio and proportion. Using their understanding of ratio and proportion as a foundation, students can then build connected and related understandings of fractions, decimals and percents. The study discussed in this paper illustrates that grounding decimal instruction in the broader context of ratio can help students gain deeper conceptual understandings of decimal notation as well as fractions and percents.     

A logarithmically completely monotonic function involving the ratio of gamma functions

In the paper, the authors concisely survey and review some functionsinvolving the gamma function and its various ratios, simply state theirlogarithmically complete monotonicity and related results, and find necessaryand sufficient conditions for a new function involving the ratio of twogamma functions and originating from the coding gain to be logarithmicallycompletely monotonic.     

Representations for the derivative at zero and finite parts of the Barnes zeta function

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes gamma function, including the particular case also known as multiple gamma function.     

The Multiple Gamma Function and Its Application to Computation of Series

The multiple gamma function Γ_n, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γ_n function and their applications to summation of series and infinite products.This work was supported by NFS grant CCR-0204003.2000 Mathematics Subject Classification: Primary—33E20, 33F99, 11M35, 11B73     

Estimates for the Approximation Numbers of One Class of Integral Operators. I

Asymptotic estimates are obtained for the approximation numbers of the Hardy integral operator with variable limits of integration.     

Completely monotone functions and the Wallis ratio

The aim of the paper is to improve known estimates of the Wallis ratio. Moreover, we show that these improvements are valid, because certain functions involving the continuous version of the Wallis ratio are completely monotone.     

A note on the rate of convergence in the strong law of large numbers for martingales

We prove a Baum-Katz-Nagaev type rate of convergence in the Marcinkiewicz-Zygmund and Kolmogorov strong laws of large numbers for norm bounded martingale difference sequences.     

Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders

In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as $x\to \infty$,$\frac{\mathrm{\Gamma }(x+p)\mathrm{\Gamma }(x+q)}{\mathrm{\Gamma }{(x+(p+q)/2)}^2}=\exp ?[\sum _{k=1}^n\frac{B_{2k}\left(s\right)?B_{2k}(1/2)}{k(2k?1){(x+r_0)}^{2k?1}}+R_n(x;p,q)]$where $p,q\in \mathbb{R}$ with $w=|p?q|\ne 0$ and $s=(1?w)/2$, $r_0=(p+q?1)/2$, $B_{2n+1}\left(s\right)$ are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function $x?{(?1)}^n{R}_n(x;p,q)$ is completely monotonic on $(?r_0,\infty )$ if $|p?q|<1$, which yields a sharp upper bound for $\left|R_n(x;p,q)\right|$. This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when $q\to p$.     

The moment function for the ratio of correlated generalized gamma variables

The moment function for the ratio of correlated generalized gamma variables is expressed in terms of special functions. The expression presented generalizes the known moment expression for the integer valued moments to the real valued moments. Approximate formulas, in terms of elementary functions, are provided for low and high correlation regions and some application examples are given.     

A Unified Derivation of the Complementary Waiting Time Distribution in Sequential Occupancy

Occupancy distributions are defined on the stochastic model of random allocation of balls to a specific number of distinguishable urns. The reduction of the joint distribution of the occupancy numbers, when a specific number of balls are allocated, to the joint conditional distribution of independent random variables given their sum, when the number of balls allocated is unspecified, is a powerful technique in the study of occupancy distributions. Consider a supply of balls randomly distributed into n distinguishable urns and assume that the number X of balls distributed into any specific urn is a random variable with probability function P(X = x) = q _x , x = 0, 1,.... The probability function of the number L _r of occupied urns until r balls are placed into previously occupied urns is derived in terms of convolutions of q _x , x = 0, 1,... and their finite differences. Further, using this distribution, the minimum variance unbiased estimator of the parameter n, based on a suitable sequential sampling scheme, is deduced. Finally, some illustrating applications are discussed.      

Generalization of the Maxwell equation and relation to the elastic equation

In this paper, we propose a hyperbolic system of first‐order pseudo‐differential equations as generalization of the Maxwell equation. We state basic properties of this system corresponding to the ones of the (usual) Maxwell equation and explain that several known generalized Maxwell equations presented by some researchers can be integrated into the system. Namely, their equations can be regarded as our equation in special cases. Their generalized equations admit not only transversal but also longitudinal waves and are examined from the physical viewpoint. Using the present system, from the mathematical viewpoint, we interpret the meaning for presence of the longitudinal wave (with the transversal one) in their generalized equations. This presence means existence of more than one non‐zero characteristic root for the system (ie, non‐zero eigenvalue of the symbol). We prove also that our system becomes a first‐order expression of (generalized) elastic equations. Furthermore, it is shown that introducing the elastic equations implies expressing the generalized Maxwell equations by the potentials.     

New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series.     

On some results of Yi and their applications in the functional equations

In this article, we establish some uniqueness theorems that improves some results of H. X. Yi for a family of meromorphic functions, and as applications, we give some results about the non-existence of meromorphic solutions of Fermat type functional equations.