algorithms
Article On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application
Kyle R. Bryenton 1,† , Andrew R. Cameron 2,†, Keegan L. A. Kirk 1,†, Nasser Saad 1,*,† , Patrick Strongman 1,† and Nikita Volodin 1,†
1 School of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada; [email protected] (K.R.B.); [email protected] or [email protected] (K.L.A.K.); [email protected] (P.S.); [email protected] (N.V.) 2 Department of Physics, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work.
Received: 24 October 2020; Accepted: 6 November 2020; Published: 9 November 2020
Abstract: The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n − 1, and n − 2 respectively. We show that for n ≥ 3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n = 4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.
Keywords: symbolic computation; algorithm; polynomial solutions; Scheffé criteria; Heun equation; Dirac equation; method of Frobenius; recurrence relations
1. Introduction Differential equations with polynomial coefficients have played an important role not only in understanding engineering and physics problems [1–32], but also as a source of inspiration for some of the most crucial results in special functions and orthogonal polynomials [33–48]. The classical differential equation