polygamma functions

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1: 5.15 Polygamma Functions

§5.15 Polygamma Functions

►The functions ${\psi }^{(n)}\left(z\right)$, $n=1,2,\mathrm{\dots }$, are called the polygamma functions. … ►

5.15.2 $${\psi }^{(n)}\left(1\right)={(-1)}^{n+1}n!\zeta \left(n+1\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), ${\psi }^{(n)}\left(z\right)$: polygamma functions and $n$: nonnegative integer

A&S Ref:

6.4.2

Referenced by:

§5.15

Permalink:

http://dlmf.nist.gov/5.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

►

5.15.3 $${\psi }^{(n)}\left(\frac{1}{2}\right)={(-1)}^{n+1}n!(2^{n+1}-1)\zeta \left(n+1\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), ${\psi }^{(n)}\left(z\right)$: polygamma functions and $n$: nonnegative integer

A&S Ref:

6.4.4

Permalink:

http://dlmf.nist.gov/5.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

… ►For $B_{2k}$ see §24.2(i). …

2: 5.21 Methods of Computation

… ►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). ►Similarly for $\ln \mathrm{\Gamma }\left(z\right)$, $\psi \left(z\right)$, and the polygamma functions. …

3: 5.16 Sums

§5.16 Sums

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4: 5.22 Tables

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§5.22(ii) Real Variables

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5: Bibliography K

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Calculation of complex polygamma functions. Sci. Rep. Hirosaki Univ. 43 (1), pp. 161–183.

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External Links:

ISSN 0367-6439, MathReview, ZentralBlatt

Referenced by:

§5.24(iv).

Encodings:

BibTeX

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6: Bibliography

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Polygamma functions of negative order. J. Comput. Appl. Math. 100 (2), pp. 191–199.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

(25.11.32), (25.11.34), §25.11(viii).

Encodings:

BibTeX

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7: Bibliography B

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Computation of the polygamma functions. Comm. Statist. B—Simulation Comput. 13 (3), pp. 409–415.

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External Links:

ISSN 0361-0918, Document, MathReview (S. Conde)

Referenced by:

§5.24(iii).

Encodings:

BibTeX

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8: Software Index

… ► ►►►

	Open Source													With Book						Commercial
…
5.24(iii) $\psi \left(x\right)$, ${\psi }^{(n)}\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓			✓		✓	✓	✓	✓	✓	✓
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

9: 5.24 Software

… ►A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C7). … ►

§5.24(iii) $\psi \left(x\right)$, ${\psi }^{(n)}\left(x\right)$, $x\in ℝ$

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§5.24(iv) $\mathrm{\Gamma }\left(z\right)$, $\psi \left(z\right)$, ${\psi }^{(n)}\left(z\right)$, $z\in ℂ$

… ►No research software has been identified for this function. …