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31: 15.19 Methods of Computation

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§15.19(iv) Recurrence Relations

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32: 28.14 Fourier Series

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28.14.4 $$q{c}_{2m+2}-\left(a-{(\nu +2m)}^2\right)c_{2m}+q{c}_{2m-2}=0,$$ $a={\lambda }_{\nu }\left(q\right),c_{2m}=c_{2m}^{\nu }(q)$,

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d), item (d)

Permalink:

http://dlmf.nist.gov/28.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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33: 26.6 Other Lattice Path Numbers

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§26.6(iii) Recurrence Relations

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26.6.10 $$D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),$$ $m,n\ge 1$,

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Symbols:

$m$: nonnegative integer, $n$: nonnegative integer and $D(m,n)$: Delannoy number

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iii), §26.6 and Ch.26

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34: 4.45 Methods of Computation

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4.45.8 $$2\arctan \frac{x}{1+{(1+x^2)}^{1/2}}=\arctan x,$$ .

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Symbols:

$\arctan z$: arctangent function and $x$: real variable

Referenced by:

Erratum (V1.0.7) for Equations (4.45.8), (4.45.9)

Permalink:

http://dlmf.nist.gov/4.45.E8

Encodings:

pMML, png, TeX

Substitution (effective with 1.0.7):

The equation that was given originally, $2\arctan \frac{{(1+x^2)}^{1/2}-1}{x}=\arctan x$, has been replaced with an equation that leads to a better recurrence for numerically computing $\arctan x$; see (4.45.9).

Suggested 2014-02-05 by Masataka Urago

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

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4.45.9 $$x_n=\frac{x_{n-1}}{1+{(1+x_{n-1}^2)}^{1/2}},$$ $n=1,2,3,\mathrm{\dots }$,

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Symbols:

$\arctan z$: arctangent function, $n$: integer and $x$: real variable

Referenced by:

(4.45.8), Erratum (V1.0.7) for Equations (4.45.8), (4.45.9)

Permalink:

http://dlmf.nist.gov/4.45.E9

Encodings:

pMML, png, TeX

Substitution (effective with 1.0.7):

The original recurrence given in this equation, $x_n=\frac{{(1+x_{n-1}^2)}^{1/2}-1}{x_{n-1}}$ is susceptible to cancellation error when $x_n$ is small. It has been replaced with another recurrence that is better for numerically computing $\arctan x$.

Suggested 2014-02-05 by Masataka Urago

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

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35: 13.29 Methods of Computation

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§13.29(iv) Recurrence Relations

►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. …

36: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

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§13.3(i) Recurrence Relations

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37: 5.17 Barnes’ $G$-Function (Double Gamma Function)

§5.17 Barnes’ $G$-Function (Double Gamma Function)

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38: 18.22 Hahn Class: Recurrence Relations and Differences

§18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(i) Recurrence Relations in $n$

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Table 18.22.1: Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials.

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$p_n(x)$	$A_n$	$C_n$
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39: Bibliography W

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

BibTeX

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

Encodings:

BibTeX

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Asymptotics of linear recurrences. Anal. Appl. (Singap.) 12 (4), pp. 463–484.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

BibTeX

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40: 8.8 Recurrence Relations and Derivatives

§8.8 Recurrence Relations and Derivatives

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