Uniform asymptotic expansions of the Jacobi polynomials and an associated function

Author:
David Elliott

Journal:
Math. Comp. **25** (1971), 309-315

MSC:
Primary 33A65

MathSciNet review:
0294737

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Abstract: Asymptotic expansions have been obtained using two theorems due to Olver for the Jacobi polynomials and an associated function. These expansions are uniformly valid for complex arguments over certain regions, for large values of the order.

**[1]**J. D. Donaldson & David Elliott,*Quadrature*II:*The Estimation of Remainders in Certain Quadrature Rules*, University of Tasmania, Mathematics Department, Technical Report No. 24, February, 1970.**[2]**A. Erdélyi, W. Magnus, D. Oberhettinger & D. G. Tricomi,*Higher Transcendental Functions*, Vol. 1, McGraw-Hill, New York, 1953. MR**15**, 419.**[3]**A. Erdélyi, W. Magnus, D. Oberhettinger & D. G. Tricomi,*Higher Transcendental Functions*, Vol. 2, McGraw-Hill, New York, 1953. MR**15**, 419.**[4]**L. M. Milne-Thomson,*The Calculus of Finite Differences*, Macmillan and Co., Ltd., London, 1951. MR**0043339****[5]**F. W. J. Olver,*The asymptotic solution of linear differential equations of the second order for large values of a parameter*, Philos. Trans. Roy. Soc. London. Ser. A.**247**(1954), 307–327. MR**0067249****[6]**F. W. J. Olver,*The asymptotic solution of linear differential equations of the second order in a domain containing one transition point*, Philos. Trans. Roy. Soc. London. Ser. A.**249**(1956), 65–97. MR**0079157****[7]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939. MR**1**, 14.**[8]**F. G. Tricomi and A. Erdélyi,*The asymptotic expansion of a ratio of gamma functions*, Pacific J. Math.**1**(1951), 133–142. MR**0043948**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1971-0294737-5

Keywords:
Uniform asymptotic expansions,
Jacobi polynomials,
hypergeometric function,
complex plane,
Olver's theorems,
modified Bessel functions

Article copyright:
© Copyright 1971
American Mathematical Society