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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 $ f(t)$, the inverse Laplace transform of $ F(p)$, where (1) $ f(t) = (1/2\pi j)\int _{c - j\infty }^{c + j\infty }{e^{{p^t}}}F(p)dp$, the function $ F(p)$ may be either known only numerically or too complicated for evaluating $ f(t)$ by Cauchy's theorem. When $ F(p)$ behaves like a polynomial without a constant term, in the variable $ 1/p$, along ( $ c - j\infty ,\;c + j\infty $), one may find $ f(t)$ numerically using new quadrature formulas (analogous to those employing the zeros of the Laguerre polynomials in the direct Laplace transform). Suitable choice of $ {p_i}$ yields an $ n$-point quadrature formula that is exact when $ {\rho _{2n}}$ is any arbitrary polynomial of the ($ 2n$)th degree in $ x \equiv 1/p$ without a constant term, namely: (2) $ (1/2\pi j)\int_{c - j\infty }^{c + j\infty } {{e^p}} {\rho _{2n}}(1/p)dp = {\sum\limits_{i = 1}^n {{A_i}} ^{(n)}}{\rho _{2n}}(1/{p_i})$. In (2), $ {x_i} \equiv 1/{p_i}$ are the zeros of the orthogonal polynomials $ {P_n}(x) \equiv \prod\limits_{i = 1}^n {(x - {x_i})}$ where (3) $ (1/2\pi j)\int_{c - j\infty }^{c + j\infty } {{e^p}} (1/p){p_n}(1/p){(1/p)^i}dp = 0,\;i = 0,1, \cdots ,n - 1$ and $ {A_i}^{(n)}$ correspond to the Christoffel numbers. The normalization $ {P_n}(1/p) \equiv (4n - 2)(4n - 6) \cdots 6{p_n}(1/p),n \geq 2$, produces all integral coefficients. $ {P_n}(1/p)$ is proven to be $ {( - 1)^n}{e^{ - p}}{p^n}{d^n}({e^p}/{p^n})/d{p^n}$. The normalization factor is proved, in three different ways, to be given by (4) $ (1/2\pi j)\int_{c - j\infty }^{c + j\infty } {{e^p}} (1/p){[{P_n}(1/p)]^2}dp = \tfrac{1}{2}{( - 1)^n}$. Proofs are given for the recurrence formula (5) $ (2n - 3){P_n}(x) = [(4n - 2)(2n - 3)x + 2]{P_{n - 1}}(x) + (2n - 1){P_{n - 2}}(x)$, for $ n \geq 3$, and the differential equation (6) $ {x^2}{P_n}^{''}(x) + (x - 1){P_n}'(x) - {n^2}{P_n}(x) = 0$. The quantities $ {p_i}^{(n)},1/{p_i}^{(n)}$ and $ {A_i}^{(n)}$ were computed, mostly to 6S - 8S, for $ i = 1(1)n,n = 1(1)8$.


References [Enhancements On Off] (What's this?)

  • [1] H. S. Carslaw & J. C. Jaeger, Operational Methods in Applied Mathematics, 2nd edition, Oxford University Press, 1949, p. 75. MR 0005988 (3:243a)
  • [2] H. E. Salzer & R. Zucker, ``Table of the Zeros and Weight Factors of the First Fifteen Laguerre Polynomials,'' Amer. Math. Soc., Bull., v. 55, 1949, p. 1004-1012. MR 0032191 (11:263m)
  • [3] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Colloquium Pub., v. 23, 1939, p. 46-47.
  • [4] H. L. Krall & O. Frink, ``A New Class of Orthogonal Polynomials: The Bessel Polynomials,'' Amer. Math. Soc., Trans., v. 65, 1, 1949, p. 100-115. MR 0028473 (10:453a)
  • [5] G. Doetsch, Theorie und Anwendung der Laplace-Transformation, Springer, Berlin, 1937, p. 128.
  • [6] The shift in notation from $ (n + 1)$ to $ n$ in $ {A_i}^{(n)}$ will cause no confusion after the $ {A_i}$'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 $ {P_n}(x) \equiv {( - 1)^n}{y_n}(x,1, - 1)$ where $ {y_n}(x,a,b)$ are ``generalized Bessel polynomials'' (see[4]).
  • [8] G. Szegö, op. cit., p. 41-42.
  • [9] Formula (14) holds for $ n = 2$ if we define $ {P_0}(x) \equiv 1$.
  • [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

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