A Uniform Expansion for the Eigenfunction of a Singular Second-Order Differential Operator
- 1 November 1990
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 21 (6) , 1619-1632
- https://doi.org/10.1137/0521088
Abstract
In a recent work, Frenzen and Wong [Canad. J. Math., 37 (1985), pp. 979–1007] have obtained a uniform asymptotic expansion for the Jacobi polynomials in terms of Bessel functions. An analogous expansion for the Jacobi functions had been given earlier by Stanton and Tomas [Acta Math., 140 (1978), pp. 251–276]. The common starting point of these papers is an integral representation. In this paper it is shown that in general such expansions can be obtained directly from an eigenfunction of a singular second-order differential operator, and with additional assumptions they converge in some interval. This leads to an expansion for the eigenfunction of an integral representation of Mehler type with good information on the kernel.Keywords
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