Polynomial solutions of differential–difference equations

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Authors

Dominici, Diego
Driver, Kathy (Cathryn Helena Stanford)
Jordaan, Kerstin Heidrun

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Publisher

Elsevier

Abstract

We investigate the zeros of polynomial solutions to the differential–difference equation Pn+1 = AnP’n + BnPn, n = 0,1,… where An and Bn are polynomials of degree at most 2 and 1 respectively. We address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlacing. Our result holds for general classes of polynomials including sequences of classical orthogonal polynomials as well as Euler–Frobenius, Bell and other polynomials.

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Keywords

Interlacing of zeros, Zeros, Euler-Frobenius polynomials, Bell polynomials

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Citation

D. Dominici, et al., Polynomial solutions of differential–difference equations, Journal of Approximation Theory, Vol. 163, no. 1, pp. 41-48(2011), doi:10.1016/j.jat.2009.05.010