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.
Highlights
IntroductionDifferential equations with polynomial coefficients have played an important role in understanding engineering and physics problems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32], and as a source of inspiration for some of the most crucial results in special functions and orthogonal polynomials [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]
We begin by stating the necessary condition for the existence of polynomial solutions to the general differential equation given by (3): Theorem 2
In Reference [55], the author claims the existence of polynomial solutions of a differential equation which matches (13) that satisfy the necessary condition (7)
Summary
Differential equations with polynomial coefficients have played an important role in understanding engineering and physics problems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32], and as a source of inspiration for some of the most crucial results in special functions and orthogonal polynomials [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. With the general theory of linear differential equations as a guide [50,51], the zeros of the leading polynomial Pn (r ) classify the solutions of the Equation (3). A different approach which depends on the parameters of the leading coefficient Pn (r ), was introduced in Reference [13] to study the polynomial solutions of (1). The criteria for polynomial solutions of second-order linear differential Equation (3) was introduced, using the Asymptotic Iteration Method, in References [52,53]: Theorem 1. The present work establishes polynomial solutions in simple and constructive forms to (3) along with all necessary and sufficient conditions they require.
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