Pochhammer%20integral

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21—30 of 545 matching pages

21: 5.18 $q$ -Gamma and $q$ -Beta Functions

… ►

5.18.1

\left(a;q\right)_{n}=\prod_{k=0}^{n-1}(1-aq^{k}),

n=0,1,2,\dots

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real or complex variable, $n$ : nonnegative integer, $k$ : nonnegative integer and $a$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(i), §5.18 and Ch.5

►

5.18.2

n!_{q}=1(1+q)\cdots(1+q+\dots+q^{n-1})=\left(q;q\right)_{n}(1-q)^{-n}.

ⓘ

Defines:: $!_{\NVar{q}}$ : $q$ -factorial (as in $n!_{q}$ )
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real or complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(i), §5.18 and Ch.5

… ►

5.18.3

\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k}).

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real or complex variable, $k$ : nonnegative integer and $a$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(i), §5.18 and Ch.5

… ►

5.18.12

\mathrm{B}_{q}\left(a,b\right)=\int_{0}^{1}\frac{t^{a-1}\left(tq;q\right)_{% \infty}}{\left(tq^{b};q\right)_{\infty}}\,{\mathrm{d}}_{q}t,

0<q<1

\Re a>0

\Re b>0

ⓘ

Symbols:: $\int$ : integral, $\mathrm{B}_{\NVar{q}}\left(\NVar{a},\NVar{b}\right)$ : $q$ -beta function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $\Re$ : real part, $q$ : real or complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(iii), §5.18 and Ch.5

►For

q

-integrals see §17.2(v).

22: 16.2 Definition and Analytic Properties

… ►

16.2.1

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum_{k% =0}^{\infty}\frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{{% \left(b_{1}\right)_{k}}\cdots{\left(b_{q}\right)_{k}}}\frac{z^{k}}{k!}.

ⓘ

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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §10.16, §10.16, §10.39, §16.2(iv), §16.2(ii), §16.2(iii), §16.2(iii), §16.2(iii), §16.5, §18.17(v), §18.17(vi), §18.17(vii), §18.20(ii), §18.26(i), §18.30(i), §18.38(ii)
Permalink:: http://dlmf.nist.gov/16.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(i), §16.2 and Ch.16

… ►

16.2.3

{{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{{\left(% \mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_{m}}}{{}_{q+1}F_{p}}% \left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q}}{z}\right)

ⓘ

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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/16.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

… ►

16.2.4

\sum_{k=0}^{m}\frac{{\left(\mathbf{a}\right)_{k}}}{{\left(\mathbf{b}\right)_{k% }}}\frac{z^{k}}{k!}=\frac{{\left(\mathbf{a}\right)_{m}}z^{m}}{{\left(\mathbf{b% }\right)_{m}}m!}{{}_{q+2}F_{p}}\left({-m,1,1-m-\mathbf{b}\atop 1-m-\mathbf{a}}% ;\frac{(-1)^{p+q+1}}{z}\right).

ⓘ

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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/16.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

… ►See §16.5 for the definition of

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

as a contour integral when

p>q+1

and none of the

a_{k}

is a nonpositive integer. … ►

16.2.5

{{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)=\ifrac{{{}_{p}F_{% q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)}{\left(\Gamma% \left(b_{1}\right)\cdots\Gamma\left(b_{q}\right)\right)}=\sum_{k=0}^{\infty}% \frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{\Gamma\left(b_{1% }+k\right)\cdots\Gamma\left(b_{q}+k\right)}\frac{z^{k}}{k!};

ⓘ

Defines:: ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.2(v)
Permalink:: http://dlmf.nist.gov/16.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(v), §16.2 and Ch.16

…

23: 19.12 Asymptotic Approximations

§19.12 Asymptotic Approximations

►With

\psi\left(x\right)

denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of

K\left(k\right)

and

E\left(k\right)

near the singularity at

k=1

is given by the following convergent series: … ►

19.12.2

E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2\right)_{m}}m!}{k^{\prime% }}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)-\frac{1}{(2m+1)(2m+% 2)}\right),

|k^{\prime}|<1

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

… ►For the asymptotic behavior of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

\phi\to\tfrac{1}{2}\pi-

and

k\to 1-

see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►Asymptotic approximations for

\Pi\left(\phi,\alpha^{2},k\right)

, with different variables, are given in Karp et al. (2007). …

24: 7.12 Asymptotic Expansions

… ►

§7.12(ii) Fresnel Integrals

►The asymptotic expansions of

C\left(z\right)

and

S\left(z\right)

are given by (7.5.3), (7.5.4), and … ►They are bounded by

|\csc\left(4\operatorname{ph}z\right)|

times the first neglected terms when

\frac{1}{8}\pi\leq|\operatorname{ph}z|<\frac{1}{4}\pi

. … ►

§7.12(iii) Goodwin–Staton Integral

►See Olver (1997b, p. 115) for an expansion of

G\left(z\right)

with bounds for the remainder for real and complex values of

z

25: 18.5 Explicit Representations

… ►

18.5.7

P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+% \beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}% \left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2% }F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.42
Referenced by:: §18.17(iv), §18.17(vii), (18.27.12_5), Table 18.3.1, §18.5(iii), §18.5(iii), §18.6(ii), §18.9(iii)
Permalink:: http://dlmf.nist.gov/18.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.11

C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda% \right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-% 2\ell)\theta\right)={\mathrm{e}}^{\mathrm{i}n\theta}\frac{{\left(\lambda\right% )_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};{\mathrm{e}}^{-2% \mathrm{i}\theta}\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 22.3.12
Referenced by:: §18.18(iv), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.12

L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1% \right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{{\left(\alpha+1\right)_{% n}}}{n!}{{}_{1}F_{1}}\left({-n\atop\alpha+1};x\right).

… ►

L^{(\alpha)}_{3}\left(x\right)=-\tfrac{1}{6}x^{3}+\tfrac{1}{2}(\alpha+3)x^{2}-% \tfrac{1}{2}{\left(\alpha+2\right)_{2}}x+\tfrac{1}{6}{\left(\alpha+1\right)_{3% }},

►

L^{(\alpha)}_{4}\left(x\right)=\tfrac{1}{24}x^{4}-\tfrac{1}{6}(\alpha+4)x^{3}+% \tfrac{1}{4}{\left(\alpha+3\right)_{2}}x^{2}-\tfrac{1}{6}{\left(\alpha+2\right% )_{3}}x+\tfrac{1}{24}{\left(\alpha+1\right)_{4}}.

…

26: 13.29 Methods of Computation

… ►

§13.29(iii) Integral Representations

►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. … ►

13.29.3

e^{-\frac{1}{2}z}=\sum_{s=0}^{\infty}\frac{{\left(2\mu\right)_{s}}{\left(\frac% {1}{2}+\mu-\kappa\right)_{s}}}{{\left(2\mu\right)_{2s}}s!}(-z)^{s}y(s),

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable and $y(n)$ : solution
Permalink:: http://dlmf.nist.gov/13.29.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

… ►

13.29.6

w(n)={\left(a\right)_{n}}U\left(n+a,b,z\right),

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.29.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

… ►

13.29.7

z^{-a}=\sum_{s=0}^{\infty}\frac{{\left(a-b+1\right)_{s}}}{s!}w(s),

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.29.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

…

27: 18.18 Sums

… ►Assume also the integrals

\int_{-1}^{1}(f(x))^{2}(1-x)^{\alpha}(1+x)^{\beta}\,\mathrm{d}x

and

\int_{-1}^{1}(f^{\prime}(x))^{2}(1-x)^{\alpha+1}(1+x)^{\beta+1}\,\mathrm{d}x

converge. … ►Assume also

\int_{0}^{\infty}(f(x))^{2}{\mathrm{e}}^{-x}x^{\alpha}\,\mathrm{d}x

converges. … ►Assume also

\int_{-\infty}^{\infty}(f(x))^{2}{\mathrm{e}}^{-x^{2}}\,\mathrm{d}x

converges. … ►For integral representations for products implied by (18.18.8) and (18.18.9) see (18.17.5) and (18.17.6), respectively. … ►

18.18.19

x^{n}={\left(\alpha+1\right)_{n}}\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell% }}}{{\left(\alpha+1\right)_{\ell}}}L^{(\alpha)}_{\ell}\left(x\right).

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18

…

28: 31.11 Expansions in Series of Hypergeometric Functions

… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►

31.11.3_1

P_{j}^{5}=\frac{{\left(\lambda\right)_{j}}{\left(1-\gamma+\lambda\right)_{j}}}% {{\left(1+\lambda-\mu\right)_{2j}}}z^{-\lambda-j}\*{{}_{2}F_{1}}\left({\lambda% +j,1-\gamma+\lambda+j\atop 1+\lambda-\mu+2j};\frac{1}{z}\right),

ⓘ

Defines:: $P_{j}^{5}$ : first Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally)
Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $\lambda$ and $\mu$
Referenced by:: §31.11(ii), Erratum (V1.1.7) for Additions
Permalink:: http://dlmf.nist.gov/31.11.E3_1
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: This equation was added.
Suggested 2022-09-14 by Hans Volkmer
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

►

31.11.3_2

P_{j}^{6}=\frac{{\left(\lambda-\mu\right)_{2j}}}{{\left(1-\mu\right)_{j}}{% \left(\gamma-\mu\right)_{j}}}z^{-\mu+j}\*{{}_{2}F_{1}}\left({\mu-j,1-\gamma+% \mu-j\atop 1-\lambda+\mu-2j};\frac{1}{z}\right).

ⓘ

Defines:: $P_{j}^{6}$ : second Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally)
Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $\lambda$ and $\mu$
Referenced by:: §31.11(ii), Erratum (V1.1.7) for Additions
Permalink:: http://dlmf.nist.gov/31.11.E3_2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: This equation was added.
Suggested 2022-09-14 by Hans Volkmer
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

… ►

31.11.12

P_{j}^{5}=\frac{{\left(\alpha\right)_{j}}{\left(1-\gamma+\alpha\right)_{j}}}{{% \left(1+\alpha-\beta+\epsilon\right)_{2j}}}z^{-\alpha-j}\*{{}_{2}F_{1}}\left({% \alpha+j,1-\gamma+\alpha+j\atop 1+\alpha-\beta+\epsilon+2j};\frac{1}{z}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $z$ : complex variable, $\gamma$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $P_{j}^{5}$ : first Kummer solution at $\infty$ of generalized hypergeometric differential equation
Referenced by:: §31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)
Permalink:: http://dlmf.nist.gov/31.11.E12
Encodings:: pMML, png, TeX
Notational Change (effective with 1.1.7):: We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.
See also:: Annotations for §31.11(iii), §31.11 and Ch.31

…

29: 16.13 Appell Functions

… ►

16.13.1

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}{\left(\beta^{% \prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},

\max\left(|x|,|y|\right)<1

ⓘ

Defines:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

►

16.13.2

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}% {\left(\beta^{\prime}\right)_{n}}}{{\left(\gamma\right)_{m}}{\left(\gamma^{% \prime}\right)_{n}}m!n!}x^{m}y^{n},

|x|+|y|<1

ⓘ

Defines:: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

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16.13.3

{F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m}}{\left(\alpha^{\prime}% \right)_{n}}{\left(\beta\right)_{m}}{\left(\beta^{\prime}\right)_{n}}}{{\left(% \gamma\right)_{m+n}}m!n!}x^{m}y^{n},

\max\left(|x|,|y|\right)<1

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Defines:: ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

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16.13.4

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m+n}}}{{\left(% \gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},

\sqrt{|x|}+\sqrt{|y|}<1

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Defines:: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

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30: 25.5 Integral Representations

§25.5 Integral Representations

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25.5.5

\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-\left\lfloor x\right\rfloor-% \frac{1}{2}}{x^{s+1}}\,\mathrm{d}x,

-1<\Re s<0

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\int$ : integral, $\Re$ : real part, $x$ : real variable and $s$ : complex variable
Keywords:: improper integral, integral representation
Source:: Titchmarsh (1986b, (2.1.6), p. 15)
Permalink:: http://dlmf.nist.gov/25.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §25.5(i), §25.5 and Ch.25

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25.5.7

\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)% !}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x% ^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\,\mathrm{d}x,

\Re s>-(2n+1)

n=1,2,3,\dots

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: improper integral, integral representation
Proof sketch:: Derivable from (25.5.6) by adding and subtracting $\sum_{m=1}^{n}B_{2m}x^{2m-1}/(2m)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that $({\mathrm{e}}^{x}-1)^{-1}-x^{-1}+1/2-\sum_{m=1}^{n}B_{2m}x^{2m-1}/(2m)!=O\left% (x^{2n+1}\right)$ as $x\to 0$ , demonstrating the region of convergence.
Referenced by:: Erratum (V1.0.9) for Chapters 7, 25
Permalink:: http://dlmf.nist.gov/25.5.E7
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.9):: We have rewritten the original summation $\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}\frac{\Gamma\left(s+2m-1\right)}{\Gamma\left% (s\right)}$ more concisely as $\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}{\left(s\right)_{2m-1}}$ using the Pochhammer symbol.
See also:: Annotations for §25.5(i), §25.5 and Ch.25

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25.5.19

\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\left(\pi s% \right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi}^{(m)}\left(1+x% \right)x^{-s}\,\mathrm{d}x,

m=1,2,3,\dots

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: improper integral, integral representation
Source:: de Bruijn (1937, p. 178)
Referenced by:: §25.5(ii)
Permalink:: http://dlmf.nist.gov/25.5.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §25.5(ii), §25.5 and Ch.25

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§25.5(iii) Contour Integrals

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