Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms

Author:
Herbert E. Salzer

Journal:
Math. Comp. **9** (1955), 164-177

MSC:
Primary 42.1X

DOI:
https://doi.org/10.1090/S0025-5718-1955-0078498-1

MathSciNet review:
0078498

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Abstract: In finding , the inverse Laplace transform of , where (1) , the function may be either known only numerically or too complicated for evaluating by Cauchy's theorem. When behaves like a polynomial without a constant term, in the variable , along ( ), one may find numerically using new quadrature formulas (analogous to those employing the zeros of the Laguerre polynomials in the direct Laplace transform). Suitable choice of yields an -point quadrature formula that is exact when is any arbitrary polynomial of the ()th degree in without a constant term, namely: (2) . In (2), are the zeros of the orthogonal polynomials where (3) and correspond to the Christoffel numbers. The normalization , produces all integral coefficients. is proven to be . The normalization factor is proved, in three different ways, to be given by (4) . Proofs are given for the recurrence formula (5) , for , and the differential equation (6) . The quantities and were computed, mostly to 6S - 8S, for .

**[1]**H. S. Carslaw and J. C. Jaeger,*Operational Methods in Applied Mathematics*, Oxford University Press, New York, 1941. MR**0005988****[2]**Herbert E. Salzer and Ruth Zucker,*Table of the zeros and weight factors of the first fifteen Laguerre polynomials*, Bull. Amer. Math. Soc.**55**(1949), 1004–1012. MR**0032191**, https://doi.org/10.1090/S0002-9904-1949-09327-8**[3]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc.,*Colloquium Pub.*, v. 23, 1939, p. 46-47.**[4]**H. L. Krall and Orrin Frink,*A new class of orthogonal polynomials: The Bessel polynomials*, Trans. Amer. Math. Soc.**65**(1949), 100–115. MR**0028473**, https://doi.org/10.1090/S0002-9947-1949-0028473-1**[5]**G. Doetsch,*Theorie und Anwendung der Laplace-Transformation*, Springer, Berlin, 1937, p. 128.**[6]**The shift in notation from to in will cause no confusion after the 's have been computed and are ready for use in (6).**[7]**It was called to the author's attention by H. L. Krall that where are ``generalized Bessel polynomials'' (see[4]).**[8]**G. Szegö,*op. cit.*, p. 41-42.**[9]**Formula (14) holds for if we define .**[10]**G. Szegö,*op. cit.*, p. 341-342.

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DOI:
https://doi.org/10.1090/S0025-5718-1955-0078498-1

Article copyright:
© Copyright 1955
American Mathematical Society