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Polynomial solutions of differential-difference equations

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Discrete Integrable Systems

We investigate the zeros of polynomial solutions to the differential-difference equation [ P_{n+1}(x)=A_{n}(x)P_{n}^{prime}(x)+B_{n}(x)P_{n}(x),~ n=0,1,dots ] where $A_n$ and $B_n$ 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 but includes sequences of classical orthogonal polynomials as well as Euler-Frobenius, Bell and other polynomials.

This talk is part of the Isaac Newton Institute Seminar Series series.

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