DLMF: Untitled Document

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15.16.1

F\left({a,b\atop c-\frac{1}{2}};z\right)F\left({c-a,c-b\atop c+\frac{1}{2}};z% \right)=\sum_{s=0}^{\infty}\frac{{\left(c\right)_{s}}}{{\left(c+\frac{1}{2}% \right)_{s}}}A_{s}z^{s},

|z|<1

,

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $A_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/15.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

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15.16.2

(1-z)^{a+b-c}F\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{\infty}A_{s}z^{s},

|z|<1

.

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $A_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/15.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

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15.16.3

F\left({a,b\atop c};z\right)F\left({a,b\atop c};\zeta\right)=\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}{\left(c-a\right)_{s}}{% \left(c-b\right)_{s}}}{{\left(c\right)_{s}}{\left(c\right)_{2s}}s!}\left(z% \zeta\right)^{s}F\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right),

|z|<1

,

|\zeta|<1

,

|z+\zeta-z\zeta|<1

.

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

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16.10.1

{{}_{p+r}F_{q+s}}\left({a_{1},\dots,a_{p},c_{1},\dots,c_{r}\atop b_{1},\dots,b% _{q},d_{1},\dots,d_{s}};z\zeta\right)=\sum_{k=0}^{\infty}\frac{{\left(\mathbf{% a}\right)_{k}}{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}(-z)^{k}}{{% \left(\mathbf{b}\right)_{k}}{\left(\gamma+k\right)_{k}}k!}{{}_{p+2}F_{q+1}}% \left({\alpha+k,\beta+k,a_{1}+k,\dots,a_{p}+k\atop\gamma+2k+1,b_{1}+k,\dots,b_% {q}+k};z\right){{}_{r+2}F_{s+2}}\left({-k,\gamma+k,c_{1},\dots,c_{r}\atop% \alpha,\beta,d_{1},\dots,d_{s}};\zeta\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, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.10 and Ch.16

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16.10.2

{{}_{p+1}F_{p}}\left({a_{1},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};z\zeta\right% )=(1-z)^{-a_{1}}\sum_{k=0}^{\infty}\frac{{\left(a_{1}\right)_{k}}}{k!}{{}_{p+1% }F_{p}}\left({-k,a_{2},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};\zeta\right)\left% (\frac{z}{z-1}\right)^{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, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.10 and Ch.16

… ►Expansions of the form

\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)

are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

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15.12.2

F\left(a,b;c;z\right)=\sum_{s=0}^{m-1}\frac{{\left(a\right)_{s}}{\left(b\right% )_{s}}}{{\left(c\right)_{s}}s!}z^{s}+O\left(c^{-m}\right),

|c|\to\infty

.

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $m$ : integer
Referenced by:: §15.19(iv)
Permalink:: http://dlmf.nist.gov/15.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(ii), §15.12 and Ch.15

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15.12.3

F\left({a,b\atop c+\lambda};z\right)\sim\frac{\Gamma\left(c+\lambda\right)}{% \Gamma\left(c-b+\lambda\right)}\sum_{s=0}^{\infty}q_{s}(z){\left(b\right)_{s}}% \lambda^{-s-b},

…

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Equation (18.38.3)

18.38.3

\sum_{m=0}^{n}P^{(\alpha,0)}_{m}\left(x\right)=\frac{{\left(\alpha+2\right)_{n% }}}{n!}{{}_{3}F_{2}}\left({-n,n+\alpha+2,\frac{1}{2}(\alpha+1)\atop\alpha+1,% \frac{1}{2}(\alpha+3)};\tfrac{1}{2}(1-x)\right)\geq 0,

x\geq-1

,

\alpha\geq-2

,

n=0,1,\dots

This equation was updated to include the value of the sum in terms of the ${{}_{3}F_{2}}$ function. Also the constraint was previously $-1\leq x\leq 1$ , $\alpha>-1$ .

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Subsection 17.7(iii)

The title of the paragraph which was previously “Andrews’ Terminating $q$ -Analog of (17.7.8)” has been changed to “Andrews’ $q$ -Analog of the Terminating Version of Watson’s ${{}_{3}F_{2}}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$ -Analog” has been changed to “Andrews’ $q$ -Analog of the Terminating Version of Whipple’s ${{}_{3}F_{2}}$ Sum (16.4.7)”.

…

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13.14.6

M_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}\sum_{s=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}}{{\left(1+2\mu\right)_{% s}}s!}z^{s}=z^{\frac{1}{2}+\mu}\sum_{n=0}^{\infty}{{}_{2}F_{1}}\left({-n,% \tfrac{1}{2}+\mu-\kappa\atop 1+2\mu};2\right)\frac{\left(-\tfrac{1}{2}z\right)% ^{n}}{n!},

2\mu\neq-1,-2,-3,\dots

,

ⓘ

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), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

…

… ►In addition to the four Appell functions there are

24

other sums of double series that cannot be expressed as a product of two

{{}_{2}F_{1}}

functions, and which satisfy pairs of linear partial differential equations of the second order. …

… ►For small values of

\|\mathbf{T}\|

the zonal polynomial expansion given by (35.8.1) can be summed numerically. … ►See Yan (1992) for the

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions of matrix argument in the case

m=2

, and Bingham et al. (1992) for Monte Carlo simulation on

\mathbf{O}(m)

applied to a generalization of the integral (35.5.8). …

… ►It can be expressed as a sum over all primes

p\leq x

: … ►Gauss and Legendre conjectured that

\pi\left(x\right)

is asymptotic to

x/\ln x

as

x\to\infty

: …(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►is the sum of the

\alpha

th powers of the divisors of

n

, where the exponent

\alpha

can be real or complex. …

… ►

15.10.8

F\left({a,b\atop n};z\right)\ln z-\sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!% {\left(1-a\right)_{k}}{\left(1-b\right)_{k}}}(-z)^{-k}\\ +\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b\right)_{k}}}{{\left(n% \right)_{k}}k!}z^{k}\left(\psi\left(a+k\right)+\psi\left(b+k\right)-\psi\left(% 1+k\right)-\psi\left(n+k\right)\right),

a,b\neq n-1,n-2,\dots,0,-1,-2,\dots

,

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $k$ : integer and $m$ : integer
A&S Ref:: 15.5.19. (Compared with (15.10.8) this result has a multiple of $F\left(a,b;n;z\right)$ added to the right-hand side. It also contains an error: the conditions on $a$ and $b$ should be $a,b\neq 1,2,\dots,m$ .)
Referenced by:: (15.10.8)
Permalink:: http://dlmf.nist.gov/15.10.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.9

F\left({-m,b\atop n};z\right)\ln z-\sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)% !{\left(m+1\right)_{k}}{\left(1-b\right)_{k}}}(-z)^{-k}+\sum_{k=0}^{m}\frac{{% \left(-m\right)_{k}}{\left(b\right)_{k}}}{{\left(n\right)_{k}}k!}z^{k}\left(% \psi\left(1+m-k\right)+\psi\left(b+k\right)-\psi\left(1+k\right)-\psi\left(n+k% \right)\right)+(-1)^{m}m!\sum_{k=m+1}^{\infty}\frac{(k-1-m)!{\left(b\right)_{k% }}}{{\left(n\right)_{k}}k!}z^{k},

a=-m

,

m=0,1,2,\dots

;

b\neq n-1,n-2,\dots,0,-1,-2,\dots

,

… ►

15.10.10

F\left({-m,-\ell\atop n};z\right)\ln z-\sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-% k-1)!{\left(m+1\right)_{k}}{\left(\ell+1\right)_{k}}}(-z)^{-k}+\sum_{k=0}^{% \ell}\frac{{\left(-m\right)_{k}}{\left(-\ell\right)_{k}}}{{\left(n\right)_{k}}% k!}z^{k}\left(\psi\left(1+m-k\right)+\psi\left(1+\ell-k\right)-\psi\left(1+k% \right)-\psi\left(n+k\right)\right)+(-1)^{\ell}\ell\,!\sum_{k=\ell+1}^{m}\frac% {(k-1-\ell)!{\left(-m\right)_{k}}}{{\left(n\right)_{k}}k!}z^{k},

a=-m

,

m=0,1,2,\dots

;

b=-\ell

,

\ell=0,1,2,\dots,m

.

…

… ►

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

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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

… ►

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

…

Gauss sum

11—20 of 68 matching pages

11: 15.16 Products

12: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

13: 15.12 Asymptotic Approximations

14: Errata

15: 13.14 Definitions and Basic Properties

16: 16.14 Partial Differential Equations

17: 35.10 Methods of Computation

18: 27.2 Functions

19: 15.10 Hypergeometric Differential Equation

20: 16.2 Definition and Analytic Properties