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11: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

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

… ►

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

… ►

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).

12: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The cofactor

A_{jk}

a_{jk}

is … ►For real-valued

a_{jk}

, … ►where

\omega_{1},\omega_{2},\dots,\omega_{n}

are the

n

th roots of unity (1.11.21). … ►If

D_{n}[a_{j,k}]

tends to a limit

L

n\to\infty

, then we say that the infinite determinant

D_{\infty}[a_{j,k}]

converges and

D_{\infty}[a_{j,k}]=L

. … ►The corresponding eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

can be chosen such that they form a complete orthonormal basis in

\mathbf{E}_{n}

. …

13: 1.12 Continued Fractions

… ►

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►

b_{0}+\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

is equivalent to

b^{\prime}_{0}+\displaystyle{\cfrac{a^{\prime}_{1}}{b^{\prime}_{1}+\cfrac{a^{% \prime}_{2}}{b^{\prime}_{2}+\cdots}}}

if there is a sequence

\{d_{n}\}^{\infty}_{n=0}

d_{0}=1

d_{n}\neq 0

, such that … ►Define … ►The continued fraction

\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

converges when … ►Then the convergents

C_{n}

satisfy …

14: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

p_{k}\left(n\right)

denotes the number of partitions of

n

into at most

k

parts. See Table 26.9.1. … ►It follows that

p_{k}\left(n\right)

also equals the number of partitions of

n

into parts that are less than or equal to

k

. ►

p_{k}\left(\leq m,n\right)

is the number of partitions of

n

into at most

k

parts, each less than or equal to

m

. …

15: 16.12 Products

… ►

16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\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, $z$ : 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:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.2

\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}};z\right)\right)^{2}={{}_{3% }F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b};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, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $z$ : 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:: §16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

ⓘ

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, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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 $n!$ ), $z$ : 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:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

…

16: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►The generalized hypergeometric function

{{}_{p}F_{q}}

with matrix argument

\mathbf{T}\in\boldsymbol{\mathcal{S}}

, numerator parameters

a_{1},\dots,a_{p}

, and denominator parameters

b_{1},\dots,b_{q}

is … ►

§35.8(iii) ${{}_{3}F_{2}}$ Case

… ►Let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. … ►Let

a_{1}+a_{2}+a_{3}+\frac{1}{2}(m+1)=b_{1}+b_{2}

; one of the

a_{j}

be a negative integer;

\Re\left(b_{1}-a_{1}\right)

\Re\left(b_{1}-a_{2}\right)

\Re\left(b_{1}-a_{3}\right)

\Re\left(b_{1}-a_{1}-a_{2}-a_{3}\right)>\frac{1}{2}(m-1)

. … ►Again, let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. …

17: 16.1 Special Notation

… ► ►►►►►

$p, q$	nonnegative integers.
…
$\rselection{a_{1},a_{2},\dots,a_{p}\\ b_{1},b_{2},\dots,b_{q}}$	real or complex parameters.
…
$\mathbf{a}$	vector $(a_{1},a_{2},\dots,a_{p})$ .
$\mathbf{b}$	vector $(b_{1},b_{2},\dots,b_{q})$ .
…

►The main functions treated in this chapter are the generalized hypergeometric function

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)

, the Appell (two-variable hypergeometric) functions

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)

{F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)

, and the Meijer

G

-function

{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)

. Alternative notations are

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

{{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)

, and

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

for the generalized hypergeometric function,

F_{1}(\alpha,\beta,\beta^{\prime};\gamma;x,y)

F_{2}(\alpha,\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y)

F_{3}(\alpha,\alpha^{\prime},\beta,\beta^{\prime};\gamma;x,y)

F_{4}(\alpha,\beta;\gamma,\gamma^{\prime};x,y)

, for the Appell functions, and

{G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)

for the Meijer

G

-function.

18: 16.18 Special Cases

… ►The

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

and

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

functions introduced in Chapters 13 and 15, as well as the more general

{{}_{p}F_{q}}

functions introduced in the present chapter, are all special cases of the Meijer

G

-function. … ►

16.18.1

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\left({% \textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_% {k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{1,p}_{p,q+1}}\left(-z;{1-a_{1},% \dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right)=\left({\textstyle\ifrac{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma% \left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}\left(-\frac{1}{z};{1,b_{1},\dots,% b_{q}\atop a_{1},\dots,a_{p}}\right).

ⓘ

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, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : 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/16.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.18 and Ch.16

►As a corollary, special cases of the

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

and

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

functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer

G

-function. …

19: 16.3 Derivatives and Contiguous Functions

… ►

16.3.1

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p% }\atop b_{1},\dots,b_{q}};z\right)=\frac{{\left(\mathbf{a}\right)_{n}}}{{\left% (\mathbf{b}\right)_{n}}}{{}_{p}F_{q}}\left({a_{1}+n,\dots,a_{p}+n\atop b_{1}+n% ,\dots,b_{q}+n};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), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : 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/16.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►

16.3.3

\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{\gamma-1}{{}_{p+1}F_% {q}}\left({\gamma,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right)={% \left(\gamma\right)_{n}}z^{\gamma+n-1}{{}_{p+1}F_{q}}\left({\gamma+n,a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{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), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : 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/16.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

►

16.3.4

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{\gamma-1}{{}_{p}F_{q+1}}% \left({a_{1},\dots,a_{p}\atop\gamma,b_{1},\dots,b_{q}};z\right)\right)={\left(% \gamma-n\right)_{n}}z^{\gamma-n-1}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop% \gamma-n,b_{1},\dots,b_{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), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : 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/16.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►Two generalized hypergeometric functions

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

are (generalized) contiguous if they have the same pair of values of

p

and

q

, and corresponding parameters differ by integers. … ►

16.3.7

{{}_{3}F_{2}}\left({a_{1}+2,a_{2},a_{3}\atop b_{1},b_{2}};z\right)a_{1}(a_{1}+% 1)(1-z)+{{}_{3}F_{2}}\left({a_{1}+1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)a_{1% }\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3}+1)\right)+{{}_{3}F_{2}}\left(% {a_{1},a_{2},a_{3}\atop b_{1},b_{2}};z\right)\left((2a_{1}-b_{1})(2a_{1}-b_{2}% )+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})\right)-{{}_{3}F_{2}}\left({a_{1}% -1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)(a_{1}-b_{1})(a_{1}-b_{2})=0.

ⓘ

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, $z$ : 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:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/16.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(ii), §16.3 and Ch.16

…

20: 3.6 Linear Difference Equations

… ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►The unwanted multiples of

g_{n}

now decay in comparison with

w_{n}

, hence are of little consequence. … ►The latter method is usually superior when the true value of

w_{0}

is zero or pathologically small. … ►beginning with

e_{0}=w_{0}

. …