# Neumann polynomial

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## 4 matching pages

##### 1: 10.23 Sums
10.23.11 $a_{k}=\frac{1}{2\pi i}\int_{|t|=c^{\prime}}f(t)O_{k}\left(t\right)\,\mathrm{d}t,$ $0,
and $O_{k}\left(t\right)$ is Neumann’s polynomial, defined by the generating function:
10.23.12 $\frac{1}{t-z}=J_{0}\left(z\right)O_{0}\left(t\right)+2\sum_{k=1}^{\infty}J_{k}% \left(z\right)O_{k}\left(t\right),$ $|z|<|t|$.
$O_{n}\left(t\right)$ is a polynomial of degree $n+1$ in $\ifrac{1}{t}:O_{0}\left(t\right)=1/t$ and
##### 2: Errata
• Equation (10.23.11)
10.23.11 $a_{k}=\frac{1}{2\pi i}\int_{|t|=c^{\prime}}f(t)O_{k}\left(t\right)\,\mathrm{d}t,$ $0

Originally the contour of integration written incorrectly as $|z|=c^{\prime}$, has been corrected to be $|t|=c^{\prime}$.

Reported by Mark Dunster on 2021-03-22

• ##### 3: 14.12 Integral Representations
14.12.5 $P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(% \mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\,\mathrm{d}t,$ $\Re\mu>0$.
###### Neumann’s Integral
14.12.13 $\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}\int_{-1}^{1}\frac{P_{n}\left(% t\right)}{x-t}\,\mathrm{d}t.$
##### 4: Bibliography S
• H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.
• I. Sh. Slavutskiĭ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.
• W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.
• O. Szász (1950) On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.
• G. Szegő (1967) Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.