Polynomial solutions of differential–difference equations
Copyright policy )Polynomial solutions of differential–difference equations
We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] 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.
- State University of New York at New Paltz United States
- University of Pretoria South Africa
- University of Cape Town (South Africa) South Africa
- University of Cape Town South Africa
- State University of New York at Potsdam United States
Bell polynomials, Mathematics(all), Numerical Analysis, Euler polynomials, Orthogonal polynomials, Applied Mathematics, Zeros of polynomial solutions of differential–difference equations, Euler-Frobenius polynomials, Polynomials, zeros of polynomial solutions of differential-difference equations, Zeros, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Euler–Frobenius polynomials, Mathematics - Classical Analysis and ODEs, Interlacing of zeros, interlacing zeros, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Analysis, 33C45, 42C05, Differential-difference equations
Bell polynomials, Mathematics(all), Numerical Analysis, Euler polynomials, Orthogonal polynomials, Applied Mathematics, Zeros of polynomial solutions of differential–difference equations, Euler-Frobenius polynomials, Polynomials, zeros of polynomial solutions of differential-difference equations, Zeros, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Euler–Frobenius polynomials, Mathematics - Classical Analysis and ODEs, Interlacing of zeros, interlacing zeros, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Analysis, 33C45, 42C05, Differential-difference equations
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