Dougall 7F6(1) sum

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1: 16.4 Argument Unity

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Rogers–Dougall Very Well-Poised Sum

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Dougall’s Very Well-Poised Sum

… ►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. … ►This is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).

2: 14.18 Sums

§14.18 Sums

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§14.18(iii) Other Sums

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Dougall’s Expansion

… ►For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …

3: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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$q$-Analog of Bailey’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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$q$-Analog of Gauss’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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$q$-Analog of Dixon’s ${}_3{}^{}F_2^{}\left(1\right)$ Sum

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F. H. Jackson’s $q$-Analog of Dougall’s ${}_7{}^{}F_6^{}\left(1\right)$ Sum

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Bailey’s Nonterminating Extension of Jackson’s ${}_8{}^{}\varphi _7^{}$ Sum

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4: 15.4 Special Cases

… ►The following results hold for principal branches when , and by analytic continuation elsewhere. … ►

§15.4(ii) Argument Unity

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Dougall’s Bilateral Sum

… ►If $a,b$ are not integers and $\mathrm{\Re }\left(c+d-a-b\right)>1$, then … ►where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when in (15.4.33) $a=-\frac{1}{3},-\frac{4}{3},-\frac{7}{3},\mathrm{\dots }$, and in (15.4.34) $a=0,-1,-2,\mathrm{\dots }$. …

5: Bibliography D

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On the computation of Mathieu functions. J. Engrg. Math. 7, pp. 39–61.

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Notes:

Variable-precision Algol program, maximum precision 15D

External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§28.36(ii), §28.36(iii).

Encodings:

BibTeX

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Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

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External Links:

ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

Encodings:

BibTeX

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

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A program to calculate internal conversion coefficients for all atomic shells without screening. Comput. Phys. Comm. 2 (7), pp. 427–432.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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6: 23 Weierstrass Elliptic and Modular
Functions

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7: 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

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8: 9.4 Maclaurin Series

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9.4.1 $$\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Ai}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Source:

Olver (1997b, p. 54)

A&S Ref:

10.4.2 (with 10.4.4 and 10.4.5)

Permalink:

http://dlmf.nist.gov/9.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

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9.4.2 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

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9.4.3 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

A&S Ref:

10.4.3 (with 10.4.4 and 10.4.5)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

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9.4.4 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

9: 26.3 Lattice Paths: Binomial Coefficients

… ►The number of lattice paths from $(0,0)$ to $(m,n)$, $m\le n$, that stay on or above the line $y=x$ is $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►

26.3.3 $$\sum _{n=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)x^n={(1+x)}^m,$$ $m=0,1,\mathrm{\dots }$,

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.1

Permalink:

http://dlmf.nist.gov/26.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

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26.3.4 $$\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)x^m=\frac{1}{{(1-x)}^{n+1}},$$ .

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.1 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

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26.3.7 $$\left(\genfrac{}{}{0.0pt}{}{m+1}{n+1}\right)=\sum _{k=n}^m\left(\genfrac{}{}{0.0pt}{}{k}{n}\right),$$ $m\ge n\ge 0$,

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iii), §26.3 and Ch.26

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26.3.10 $$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)=\sum _{k=0}^n{(-1)}^{n-k}\left(\genfrac{}{}{0.0pt}{}{m+1}{k}\right),$$ $m\ge n\ge 0$,

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iv), §26.3 and Ch.26

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10: 26.13 Permutations: Cycle Notation

… ► ${𝔖}_n$ denotes the set of permutations of $\{1,2,\mathrm{\dots },n\}$. … ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.