Nasser Saad et al 2006 J. Phys. A: Math. Gen. 39 13445 doi:10.1088/0305-4470/39/43/004
Nasser Saad1, Richard L Hall2 and Hakan Ciftci3
Show affiliationsWe 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.
02.30.Hq Ordinary differential equations
Issue 43 (27 October 2006)
Received 21 July 2006, in final form 11 September 2006
Published 11 October 2006
Nasser Saad et al 2006 J. Phys. A: Math. Gen. 39 13445
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