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Orthogonal Polynomials Arising in the Numerical Evaluation of Inverse Laplace Transforms

Herbert E. Salzer
Mathematical Tables and Other Aids to Computation
Vol. 9, No. 52 (Oct., 1955), pp. 164-177
DOI: 10.2307/2002053
Stable URL: http://www.jstor.org/stable/2002053
Page Count: 14
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Orthogonal Polynomials Arising in the Numerical Evaluation of Inverse Laplace Transforms
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Abstract

In finding f(t), the inverse LAPLACE transform of F(p), where (1) f(t) = (1/2π j) ∫c+j∞c-j∞ eptF(p)dp, the function F(p) may be either known only numerical 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∞, c + j∞), 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 pi yields an n-point quadrature formula that is exact when ρ2n is any arbitrary polynomial of the (2n)th degree in $x \equiv 1/p$ without a constant term, namely: (2) (1/2π j) ∫c+j∞c-j∞ ep ρ2n(1/p)dp = ∑i=1n Ai(n) ρ2n(1/pi). In (2), $x_i \equiv 1/p_i$ are the zeros of the orthogonal polynomials $p_n(x) \equiv 1/p_i$ are the zeros of the orthogonal polynomials $p_n(x) \equiv \Pi^n_{i=1} (x - x_i)$ where (3) (1/2π j) ∫c+j∞c-j∞ ep(1/p)pn(1/p)(1/p)idp = 0, i = 0, 1, ⋯, n - 1 and Ai(n) correspond to the CHRISTOFFEL numbers. The normalization $P_n(1/p) \equiv (4n - 2)(4n - 6) \cdots 6p_n(1/p), n \geq 2$, produces all integral coeffients. Pn(1/p) is proven to be (-1)n e-ppndn(ep/pn)/dpn. The normalization factor is proved, in three different ways, to be given by (4) (1/2π j) ∫c+j∞c-j∞ ep(1/p)[ Pn(1/p) ]2dp = 1/2(-1)n. Proofs are given for the recurrence formula (5) (2n - 3)Pn(x) = [ (4n - 2)(2n - 3)x + 2 ] Pn-1(x) + (2n - 1)Pn-2(x), for n ≥ 3, and the differential equation (6) x2Pn"(x) + (x - 1)Pn'(x) - n2Pn(x) = 0. The quantities pi(n), 1/pi(n) and Ai(n) were computed, mostly to 6S - 8S, for i = 1(1)n, n = 1(1)8.

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