Neumann polynomial

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1: 10.23 Sums

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10.23.11 $$a_k=\frac{1}{2\pi \mathrm{i}}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$ ,

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Defines:

$a_k$ (locally)

Symbols:

$O_n\left(x\right)$: Neumann’s polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: nonnegative integer, $z$: complex variable and $f(t)$

A&S Ref:

9.1.83

Referenced by:

Erratum (V1.1.2) for Equation (10.23.11)

Permalink:

http://dlmf.nist.gov/10.23.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

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

Reported 2021-03-22 by Mark Dunster

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

►and $O_k\left(t\right)$ is Neumann’s polynomial, defined by the generating function: ►

10.23.12 $$\frac{1}{t-z}=J_0\left(z\right)O_0\left(t\right)+2\sum _{k=1}^{\mathrm{\infty }}{J}_k\left(z\right)O_k\left(t\right),$$ .

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Defines:

$O_n\left(x\right)$: Neumann’s polynomial

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $n$: integer, $k$: nonnegative integer, $x$: real variable and $z$: complex variable

A&S Ref:

9.1.84

Permalink:

http://dlmf.nist.gov/10.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

► $O_n\left(t\right)$ is a polynomial of degree $n+1$ in $1/t:O_0\left(t\right)=1/t$ and ►

10.23.13 $$O_n\left(t\right)=\frac{1}{4}\sum _{k=0}^{\lfloor n/2\rfloor }\frac{(n-k-1)!n}{k!}{\left(\frac{2}{t}\right)}^{n-2k+1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$O_n\left(x\right)$: Neumann’s polynomial, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $n$: integer and $k$: nonnegative integer

A&S Ref:

9.1.85

Referenced by:

§10.23(iii)

Permalink:

http://dlmf.nist.gov/10.23.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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2: Errata

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10.23.11 $$a_k=\frac{1}{2\pi i}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$

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

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3: 14.12 Integral Representations

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14.12.5 $$P_{\nu }^{-\mu }\left(x\right)=\frac{{\left(x^2-1\right)}^{-\mu /2}}{\mathrm{\Gamma }\left(\mu \right)}{\int }_1^x{P}_{\nu }\left(t\right){(x-t)}^{\mu -1}dt,$$ $\mathrm{\Re }\mu >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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Neumann’s Integral

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14.12.13 $${𝑸}_n\left(x\right)=\frac{1}{2(n!)}{\int }_{-1}^1\frac{P_n\left(t\right)}{x-t}dt.$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

8.8.3 (modified)

Permalink:

http://dlmf.nist.gov/14.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14

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4: Bibliography S

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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External Links:

ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:

ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

BibTeX

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Szegő polynomials from hypergeometric functions. Proc. Amer. Math. Soc. 138 (12), pp. 4259–4270.

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External Links:

ISSN 0002-9939, Link, MathReview (Andrei Martínez Finkelshtein), ZentralBlatt

Referenced by:

§18.33(iv).

Encodings:

BibTeX

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Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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External Links:

Abstract

Referenced by:

§18.16(v).

Encodings:

BibTeX

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On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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