# Differential and Difference Equations for Products of Classical Orthogonal Polynomials

- 646 Downloads

## Abstract

Factorization of differential equations has been intensively studied (see, for instance, [1], [2]). Less results are known for difference equations. In this publication we are not giving a general approach to the theory of factorization but rather present some observations and derive formulae for further use in reference books and for symbolic computations.

Several specific examples which arise from the theory of classsical orthogonal polynomials are studied. They have, to our mind, significance for practical applications in physics.

The paper is based to some extent on the ideas developed in other publications of the authors [3], [5], [6] but the angle of view on the problem is different. In the first section differential equations are dealt with. In the second section, our studies are concentrated on difference equations. In both cases knowing the equation for orthogonal polynomials we derive equations for their products. These latter equations are of higher order than the starting ones, and polynomial solutions can be sought as solutions of multiparametric spectral problem [4].

## Keywords

Orthogonal Polynomial Legendre Polynomial Shift Operator Hermite Polynomial Jacobi Polynomial## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Berkovich, L.M.: Factorization and transformation of differential equations. R&C Dynamics (2002) (in Russian)Google Scholar
- 2.Schwartz, F.: A factorization algorithm for linear ordinary equations. In: Proc. ACM-SIGSAM (1989)Google Scholar
- 3.Slavyanov, S.Y.: The equation for the product of solutions of two different Schrödinger equations. Theor. & Math. Phys. 136, 1251–1257 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
- 4.Volkmer, H.: Multiparameter eigenvalue problem and expansion theorems. Springer, Heidelberg (1988)Google Scholar
- 5.Slavyanov, S.Y., Papshev, V.Y.: Product of the Jacobi polynomials. Ulmer Seminare 8, 346–352 (2003)Google Scholar
- 6.Slavyanov, S.Y., Papshev, V.Y.: Explicit and asymptotic study of one-dimensional multipole matrix elements. In: Nonadiabatic Transitions in Quantum Systems, Chernogolovka, pp. 84–93 (2004)Google Scholar
- 7.Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. NIST (1964)Google Scholar