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Strong stability preserving (SSP) coefficients govern the maximally allowable step-size at which positivity or contractivity preservation of integration methods for initial value problems is guaranteed. In this paper, we show that the task of computing optimal linear SSP coefficients of explicit one-step methods is, to a certain extent, equivalent to the problem of characterizing positive quadratures with integer nodes with respect to Poisson–Charlier measures. Using this equivalence, we provide sharp upper and lower bounds for the optimal linear SSP coefficients in terms of the zeros of generalized Laguerre orthogonal polynomials. This in particular provides us with a sharp upper bound for the optimal SSP coefficients of explicit Runge–Kutta methods. Also based on this equivalence, we propose a highly efficient and stable algorithm for computing these coefficients, and their associated optimal linear SSP methods, based on adaptive spectral transformations of Poisson–Charlier measures. The algorithm possesses the remarkable property that its complexity depends only on the order of the method and thus is independent of the number of stages. Our results are achieved by adapting and extending an ingenious technique by Bernstein (Acta Math 52:1–66, 1928) in his seminal work on absolutely monotonic functions. Moreover, the techniques introduced in this work can be adapted to solve the integer quadrature problem for any positive discrete multi-parametric measure supported on N under some mild conditions on the zeros of the associated orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present linear differential equations for the generating functions of the Poisson-Charlier, actuarial, and Meixner polynomials. Also, we give an application for each case.

The aim of the paper is to study the characterization of delta operator associated with some Sheffer polynomials. In this paper, we consider Poisson-Charlier polynomials and investigate the characterization of delta operator via sequential representation of delta operator. From our investigation, we are able to prove an interesting propositions for the above mentioned. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson−Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

In this paper, we present linear differential equations for the generating functions of the Poisson-Charlier, actuarial, and Meixner polynomials. Also, we give an application for each case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Barnes-type Narumi of the second kind and Poisson-Charlier mixed-type polynomials. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials. [ABSTRACT FROM AUTHOR]

Rényi's thinning operation on a discrete random variable is a natural discrete analog of the scaling operation for continuous random variables. The properties of thinning are investigated in an information-theoretic context, especially in connection with information-theoretic inequalities related to Poisson approximation results. The classical Binomial-to-Poisson convergence (sometimes referred to as the “law of small numbers”) is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is provided for this limit, and nonasymptotic bounds are also established. This development parallels, in part, the development of Gaussian inequalities leading to the information-theoretic version of the central limit theorem. In particular, a “thinning Markov chain” is introduced, and it is shown to play a role analogous to that of the Ornstein-Uhlenbeck process in connection to the entropy power inequality. [ABSTRACT FROM PUBLISHER]

This paper deals with the hypergeometric orthogonal polynomials with special emphasis on the Hahn polynomials and the continuous dual Hahn polynomials. New representations, generating functions, and summation formulas are derived. The addition theorems for the Krawtchouk and the Meixner-Pollaczek polynomials are also included. [ABSTRACT FROM AUTHOR]

A new combinatorial formalism is developed and applied to obtain new forms for Krawtchouk and Poisson-Charlier polynomials. These forms are used to obtain linear representations for pairwise products of such polynomials. As another application a family of polynomials orthogonal with respect to the multinomial distribution is established. [ABSTRACT FROM AUTHOR]

Let us choose a positive integer n and denote F(x, y)= $$\sum _{m = 0}^n f^{(n - m)} (x)g^{(m)} (y)$$ , where f(·) and g(·) are arbitrary sufficiently smooth functions. Three different proofs of the validity of the relation are given. We also establish discrete and noncommutative analogs of this identity. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. [ABSTRACT FROM AUTHOR]