A note on generating functions and summation formulae for Meixner polynomials of several variables
Abstract The present paper deals with certain generating functions and various elegant summation formulae for Meixner polynomials of several variables.
Some families of generating functions associated with the Stirling numbers of the second kind
The object of this paper is to present a systematic introduction to (and several interesting applications of) a general result on generating functions (associated with the Stirling numbers of the second kind) for a fairly wide variety of special functions and polynomials in one, two, and more variables. The main results given below are shown to apply not only to the classical orthogonal polynomials including, for example, the Jacobi polynomials (which contain, as their special cases, the Gegenbauer or ultraspherical polynomials, the Legendre or spherical polynomials, and the Chebyshev polynomials of the first and second kinds) and the Laguerre polynomials, and to their various extensions and generalizations studied in recent years, but indeed also to a class of generalized hypergeometric functions, the Lauricella polynomials in several variables, and the familiar Lagrange polynomials which arise in certain problems in statistics. Relevant connections of some of these families of generating functions with various known results are also indicated.
An acceleration method for the power series of entire functions of order 1
When f(z) is given by a known power series expansion, it is possible to construct the power series expansion for f(z; p) = e/sup -p/z f(z). We define p/sub opt/ to be the value of p for which the expansion for f(z; p) converges most rapidly. When f(z) is an entire function of order 1, we show that p/sub opt/ is uniquely defined and may be characterized in terms of the set of singularities z/sub i/ = 1/sigma/sub i/ of an associated function h(z). Specifically, it is the center of the smallest circle in the complex plane which contains all points sigma/sub i/.