Criterion for polynomial solutions to a class of linear differential equation of second order

Abstract

We consider the differential equations y''=λ_0(x)y'+s_0(x)y, where λ_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then δ_n=λ_n s_{n-1}-λ_{n-1}s_n=0, where λ_{n}= λ_{n-1}^\prime+s_{n-1}+λ_0λ_{n-1}\hbox{and}\quad s_{n}=s_{n-1}^\prime+s_0λ_{k-1},\quad n=1,2,.... Conversely (ii) if λ_nλ_{n-1}\ne 0 and δ_n=0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.

12 pages

References in corpus (6)

• Asymptotic iteration method for eigenvalue problems
• Construction of exact solutions to eigenvalue problems by the asymptotic iteration method
• Perturbation theory in a framework of iteration methods
• Arbitrary l-state solutions of the rotating Morse potential by the asymptotic iteration method
• Sextic anharmonic oscillators and orthogonal polynomials
• On polynomial solutions of differential equations

Cited by in corpus (18)

• The Klein-Gordon equation with the Kratzer potential in d dimensions
• Physical applications of second-order linear differential equations that admit polynomial solutions
• Exact and approximate solutions of Schroedinger's equation for a class of trigonometric potentials
• An Improvement of the Asymptotic Iteration Method for Exactly Solvable Eigenvalue Problems
• The Asymptotic Iteration Method Revisited
• Exact and approximate solutions to Schroedinger's equation with decatic potentials
• Solutions for certain classes of Riccati differential equation
• Exact and approximate solutions of Schrödinger's equation with hyperbolic double-well potentials
• Analytical approximate bound state solution of Schrödinger equation in $D$-dimensions with a new mixed class of potential for arbitrary $\ell$-state via asymptotic iteration method
• Discrete spectra for confined and unconfined -a/r + b r^2 potentials in d dimensions
• Soft and hard confinement of a two-electron quantum system
• On polynomial solutions to Fokker-Planck and sinked density evolution equations
• Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models
• Iterative solution of differential equations
• Spectra generated by a confined softcore Coulomb potential
• Solvable Systems of Linear Differential Equations
• A discrete and $q$ Asymptotic Iteration Method
• On Polynomial Solutions of Linear Differential Equations with Applications