A generalization of Laguerre polynomials. (English) Zbl 0780.33007

Koekoek, R.; Meijer, H. G.

SIAM J. Math. Anal. 24, No. 3, 768-782 (1993).

The authors investigate orthogonal polynomials for the inner product \[ \langle p,q\rangle= \int_ 0^ \infty {x^ \alpha e^{-x}\over \Gamma(\alpha+1)} p(x)q(x) dx+Mp(0)q(0) + Np'(0) q'(0), \] thereby generalizing the Laguerre polynomials \((M=N=0)\) and Koornwinder’s Laguerre-type polynomials \((N=0)\). For these generalized Laguerre polynomials one obtains a second order differential equation, a five-term recurrence relation, a Christoffel-Darboux type formula and a representation as a \({_ 3F_ 3}\)-hypergeometric series. It is shown that the polynomial of degree \(n\) has \(n\) real and simple zeros and at most one zero is negative, in which case a lower bound is obtained for this zero.

Reviewer: W.Van Assche (Heverlee)

MSC:

33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Keywords:

Sobolev inner product; Laguerre polynomials; zero