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

Author:
Nasser Saad   Richard L Hall and Hakan Ciftci  


Journal:
Journal of Physics A: General Physics


Issue Date:
2006


Abstract(summary):

We consider the differential equations y″ = λ0(x)y' + s0(x)y, where λ0(x), s0(x) are C-functions. We prove (i) if the differential equation has a polynomial solution of degree n > 0, then δn = λnsn-1 - λn-1sn = 0, where λn = λ'n-1 + sn-1 + λ0λn-1andsn = s'n-1 + s0λk-1, n = 1, 2, .... Conversely (ii) if λnλn-1 ≠ 0 and δn = 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 kinds), 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.


Page:
13445-13454


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