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 some orthogonal polynomial integrals
The modified moments of the weight functions w(x) = x (1 — x) ln(l/x), on [0, 11, with respect to the shifted Jacobi polynomials Pn^a^\x) = P^^(2x 1), and w (x) = x e (lnx) , p = 1, 2, on [0, °°), with respect to the generalized La(a) guerre polynomials Lyn '(x), are explicitly evaluated. 1. A Jacobi Polynomial Integral. In a recent paper, Gautschi [3], generalizing a result of Blue [2], has considered and explicitly evaluated the modified moments of the weight function w(x) = x"ln(l/x), p>-l, on [0, 1], with respect to the shifted Legendre polynomials Pjfrx) = PA[2x 1). We further generalize these results by considering the weight function (1.1) w(x) = xp(l-xfln(l/x), a, p>-l, and evaluating its modified moments on [0, 1] with respect to the shifted Jacobi polynomials Pfa'ß\x) = Pna'ß\2x 1). It is convenient from now on to replace p by ß + p; thus, the modified moments we have to examine assume the form (1.2) ^(P) = ío xß + ßV xTHUx)P^ß\x)dx, a,ß,ß + p>-l,n = 0,l,2,... . We easily see that vna'e\p) = -2-(a+ß+»+i) £ (1 tro + ry+Mln(7Í(l + t))P^a-ß\t)dt = _2-(a+/3+u+i)jJ^ (l-0a(l +^+"ln(l + t)p(na-ß\t)dt -In 2 • Ç_x (1 tfil + tf+>*Pn"-ß\t)dt \ ,