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.
A Class of Generating Functions
Generating functions of the type $\sum_0^\infty {L_n^{(\alpha + \beta x)} (x)z^n } $, $P_n^{(\alpha + \gamma n,\beta + \delta n)} (x)z^n $ have been obtained recently. In the present paper a general result of this kind is derived making use of the Lagrange expansion. A number of applications are given. These include, in addition to the Laguerre and Jacobi polynomials, the Hermite, Legendre, Bernoulli, Euler and Bell polynomials.
On two methods for accelerating convergence of series
SummaryThe Euler-Knopp transformation and a recently considered transformation, effective for entire function of order 1, are applied to series involving completely monotonic coefficients. Some properties of the resulting series are analyzed; these include uniform convergence with respect to the index, “a priori” and “a posteriori” estimates of the remainder. For the latter transformation a compact recursive algorithm is established which enables one to make effective use of the transformation. To illustrate the effectiveness of the transformations three applications, with examples, are included.