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On a differential equation for Koornwinder's generalized Laguerre polynomials

1991
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Proceedings of the American Mathematical Society
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Koornwinder's generalized Laguerre polynomials {L°' (x)}^=0 are orthogonal on the interval [0, oo) with respect to the weight function r,'+|.xae~j: + Nô(x), a > -1 , N > 0. We show that these polynomials for N > 0 satisfy a unique differential equation of the form oo NY^aAx)y{'\x) + xy"(x) + (a+ 1 -x)y'(x) + ny(x) = 0, 1=0 where {a,-(Ar)},^0 are continuous functions on the real line and {ai(x)}'?x are independent of the degree n . If N > 0 , only in the case of nonnegative integer values of a this differential equation is of finite order.

doi:10.1090/s0002-9939-1991-1047003-9
fatcat:32l222mwkjfmpasmlel7rtxqye