Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms

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 $i = 1(1)n,n = 1(1)8$.