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

@inproceedings{Saad2006CriterionFP,
  title={Criterion for polynomial solutions to a class of linear differential equations of second order},
  author={Nasser Saad and Richard L. Hall and Hakan Çiftçi},
  year={2006}
}
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… CONTINUE READING

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