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A generalization of Laguerre polynomials. (English) Zbl 0780.33007
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.

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
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