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11: 8.2 Definitions and Basic Properties

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P\left(a,z\right)=\frac{\gamma\left(a,z\right)}{\Gamma\left(a\right)},

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Q\left(a,z\right)=\frac{\Gamma\left(a,z\right)}{\Gamma\left(a\right)},

… ►When

z\neq 0

\Gamma\left(a,z\right)

is an entire function of

a

, and

\gamma\left(a,z\right)

is meromorphic with simple poles at

a=-n

n=0,1,2,\dots

, with residue

(-1)^{n}/n!

. … ►

8.2.12

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(1+\frac{1-a}{z}\right)\frac{% \mathrm{d}w}{\mathrm{d}z}=0.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/8.2.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §8.2(iii), §8.2 and Ch.8

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8.2.13

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(1+\frac{1-a}{z}\right)\frac{% \mathrm{d}w}{\mathrm{d}z}+\frac{1-a}{z^{2}}w=0.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/8.2.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §8.2(iii), §8.2 and Ch.8

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12: 26.6 Other Lattice Path Numbers

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Table 26.6.3: Narayana numbers

N(n,k)

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$n$	$k$
$n$	…
10	0	1	45	540	2520	5292	5292	2520	540	45	1

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13: Bibliography E

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E. B. Elliott (1903) A formula including Legendre’s

EK^{\prime}+KE^{\prime}-KK^{\prime}=\frac{1}{2}\pi

. Messenger of Math. 33, pp. 31–32.

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Notes:: Errata on p. 45 of the same volume.
External Links:: ZentralBlatt
Referenced by:: §15.16.
Encodings:: BibTeX

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W. N. Everitt (2005b) Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950. In Sturm-Liouville theory, pp. 45–74.

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External Links:: Document, Link, MathReview Entry
Referenced by:: §1.18(ix), §1.18(x).
Encodings:: BibTeX

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14: 3.4 Differentiation

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B_{-3}^{6}=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),

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B_{3}^{6}=\tfrac{1}{720}(12+8t-45t^{2}-20t^{3}+15t^{4}+6t^{5}).

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B_{-1}^{7}=-\tfrac{1}{240}(144-360t-48t^{2}+260t^{3}-45t^{4}-30t^{5}+7t^{6}),

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15: 22.18 Mathematical Applications

… ►With the identification

x=\operatorname{sn}\left(z,k\right)

y=\ifrac{\mathrm{d}(\operatorname{sn}\left(z,k\right))}{\mathrm{d}z}

, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …

16: 34.2 Definition: $\mathit{3j}$ Symbol

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34.2.4

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3% })\left((j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!% (j_{3}-m_{3})!\right)^{\frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_% {3}-s)!(j_{1}-m_{1}-s)!(j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2% }+s)!},

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Defines:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol
Symbols:: $!$ : factorial (as in $n!$ ), $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $\Delta(j_{1}j_{2}j_{3})$ : product and $s$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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34.2.5

\Delta(j_{1}j_{2}j_{3})=\left(\frac{(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-% j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}\right)^{\frac{1}{2}},

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Defines:: $\Delta(j_{1}j_{2}j_{3})$ : product (locally)
Symbols:: $!$ : factorial (as in $n!$ ) and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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34.2.6

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-% 1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1};1\right)},

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

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17: Bibliography L

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J. Lepowsky and R. L. Wilson (1982) A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.16.
Encodings:: BibTeX

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S. Lewanowicz (1985) Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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External Links:: ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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S. Lewanowicz (1987) Corrigenda: “Recurrence relations for hypergeometric functions of unit argument” [Math. Comp. 45 (1985), no. 172, 521–535; MR 86m:33004]. Math. Comp. 48 (178), pp. 853.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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18: Bibliography Z

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H. S. Zuckerman (1939) The computation of the smaller coefficients of

J(\tau)

. Bull. Amer. Math. Soc. 45 (12), pp. 917–919.

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External Links:: FullText, MathReview (H. Rademacher), ZentralBlatt
Referenced by:: §23.17(ii).
Encodings:: BibTeX

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19: 10.67 Asymptotic Expansions for Large Argument

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10.67.11

\operatorname{ber}x\operatorname{ber}'x+\operatorname{bei}x\operatorname{bei}'% x\sim\frac{e^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}-\frac{3}{8}\frac{1}{% x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{45}{512}\frac{1}{x^{3}}+\frac{31% 5}{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
A&S Ref:: 9.10.29
Permalink:: http://dlmf.nist.gov/10.67.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

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10.67.15

\operatorname{ker}x\operatorname{ker}'x+\operatorname{kei}x\operatorname{kei}'% x\sim-\frac{\pi}{2x}e^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}% {x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{45}{512}\frac{1}{x^{3}}+\frac{3% 15}{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
A&S Ref:: 9.10.33
Permalink:: http://dlmf.nist.gov/10.67.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

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20: Bibliography G

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W. Gautschi (1996) Orthogonal Polynomials: Applications and Computation. In Acta Numerica, 1996, A. Iserles (Ed.), Acta Numerica, Vol. 5, pp. 45–119.

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External Links:: MathReview (Sven Ehrich), ZentralBlatt
Referenced by:: §3.5(v).
Encodings:: BibTeX

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E. T. Goodwin (1949b) The evaluation of integrals of the form

\int^{\infty}_{-\infty}f(x)e^{-x^{2}}dx

. Proc. Cambridge Philos. Soc. 45 (2), pp. 241–245.

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External Links:: MathReview (S. Levy), ZentralBlatt
Referenced by:: §3.5(i).
Encodings:: BibTeX

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D. Goss (1978) Von Staudt for

{\mathbf{F}}_{q}[{T}]

. Duke Math. J. 45 (4), pp. 885–910.

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External Links:: ISSN 0012-7094, Document, MathReview (Neal Koblitz), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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