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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.

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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