On polynomial solutions of linear differential equations with polynomial coefficients

1977 ◽  
Vol 45 (4) ◽  
pp. 358-360 ◽  
Author(s):  
Do Tan Si
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Polynomial Solutions ◽  
Polynomial Coefficients ◽  
Linear Differential

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

Algorithms ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 286
Author(s):  
Kyle R. Bryenton ◽  
Andrew R. Cameron ◽  
Keegan L. A. Kirk ◽  
Nasser Saad ◽  
Patrick Strongman ◽  
...  
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Differential Equations ◽  
Theoretical Physics ◽  
Necessary Condition ◽  
Polynomial Solutions ◽  
Polynomial Coefficients ◽  
Linear Differential ◽  
Necessary And Sufficient

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.

A Recursion Formula for the Coefficients of Entire Functions Satisfying an ODE with Polynomial Coefficients

2004 ◽  
Vol 11 (3) ◽  
pp. 409-414
Author(s):  
C. Belingeri
Keyword(s):  
Differential Equations ◽  
Sum Rules ◽  
Entire Functions ◽  
Recursion Formula ◽  
Linear Differential Equations ◽  
Bell Polynomials ◽  
Polynomial Coefficients ◽  
Linear Differential

Abstract A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials.

scholarly journals Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yūki Naito ◽  
Mervan Pašić
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Zero Solution ◽  
Nonautonomous System ◽  
Linear Differential Equations ◽  
Solution Curve ◽  
Nonautonomous Systems ◽  
Polynomial Coefficients ◽  
The Matrix ◽  
Linear Differential

We study a new kind of asymptotic behaviour near for the nonautonomous system of two linear differential equations: , , where the matrix-valued function has a kind of singularity at . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that as and the length of the solution curve of is finite (resp., infinite) for every . It is characterized in terms of certain asymptotic behaviour of the eigenvalues of near . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at .

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