On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

• Main Authors: Kyle R. Bryenton, Andrew R. Cameron, Keegan L. A. Kirk, Nasser Saad, Patrick Strongman, Nikita Volodin
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Algorithms
• Subjects: symbolic computation; algorithm; polynomial solutions; Scheffé criteria; Heun equation; Dirac equation
• Online Access: https://www.mdpi.com/1999-4893/13/11/286

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 <i>n</i>, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> respectively. We show that for <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula> 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 <i>n</i> 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 <i>n</i>, 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 <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>. 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.