hypergeometric equation

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21: 31.10 Integral Equations and Representations

… ►

31.10.9

\mathcal{K}(\theta,\phi)=P\begin{Bmatrix}0&1&\infty&\\[1.0pt] 0&\frac{1}{2}-\delta-\sigma&\alpha&{\cos}^{2}\theta\\[1.0pt] 1-\gamma&\frac{1}{2}-\epsilon+\sigma&\beta&\end{Bmatrix}\*P\begin{Bmatrix}0&1&% \infty&\\[1.0pt] 0&0&-\frac{1}{2}+\delta+\sigma&{\cos}^{2}\phi\\[1.0pt] 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix},

… ►

31.10.11

\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\left(\ifrac{zt}{a}\right)^% {-\frac{1}{2}+\delta+\sigma-\alpha}\*{{}_{2}F_{1}}\left({\frac{1}{2}-\delta-% \sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-\gamma\atop\alpha-\beta+1};% \frac{a}{zt}\right)\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.

ⓘ

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, $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel and $\sigma$ : separation constant
Permalink:: http://dlmf.nist.gov/31.10.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

… ►This equation can be solved in terms of hypergeometric functions (§15.11(i)): ►

31.10.22

\mathcal{K}(r,\theta,\phi)=r^{m}{\sin}^{2p}\theta P\begin{Bmatrix}0&1&\infty&% \\ 0&0&a&{\cos}^{2}\theta\\ \tfrac{1}{2}(3-\gamma)&c&b&\end{Bmatrix}\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&a^{\prime}&{\cos}^{2}\phi\\ 1-\epsilon&1-\delta&b^{\prime}&\end{Bmatrix},

…

22: 15.2 Definitions and Analytical Properties

… ►

15.2.3_5

\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}% \left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1% )!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),

n=0,1,2,\dots

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.2(ii), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)
Permalink:: http://dlmf.nist.gov/15.2.E3_5
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.28):: This equation was given a number.
See also:: Annotations for §15.2(ii), §15.2 and Ch.15

… ►(Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does

\mathbf{F}\left(a,b;c;z\right)

, which is analytic at

c=0,-1,-2,\dots

.) …

23: 17.6 ${{}_{2}\phi_{1}}$ Function

… ►

17.6.1

{{}_{2}\phi_{1}}\left({a,b\atop c};q,\ifrac{c}{(ab)}\right)=\frac{\left(c/a,c/% b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty}},

|c|<|ab|

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.2.0) for Equation (17.6.1)
Permalink:: http://dlmf.nist.gov/17.6.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This constraint $|c|<|ab|$ was added.
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

17.6.4_5

{{}_{2}\phi_{1}}\left({b^{2},\ifrac{b^{2}}{c}\atop cq^{2}};q^{2},\ifrac{cq^{3}% }{b^{2}}\right)=\frac{1}{2b}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(% cq^{2},\ifrac{cq}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{cq}{b}% ;q\right)_{\infty}}{\left(b;q\right)_{\infty}}-\frac{\left(\ifrac{-cq}{b};q% \right)_{\infty}}{\left(-b;q\right)_{\infty}}\right),

\left|cq^{3}\right|<\left|b^{2}\right|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Source:: Verma and Jain (1983, (3.6))
Referenced by:: §17.6(i), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/17.6.E4_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

17.6.16

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,c/a,az,q/(az);q% \right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a% ,aq/c\atop aq/b};q,cq/(abz)\right)+\frac{\left(a,c/b,bz,q/(bz);q\right)_{% \infty}}{\left(c,a/b,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({b,bq/c% \atop bq/a};q,cq/(abz)\right),

|z|<1

|cq|<|abz|

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Referenced by:: §17.6(ii), Erratum (V1.1.9) for Equation (17.6.16)
Permalink:: http://dlmf.nist.gov/17.6.E16
Encodings:: pMML, png, TeX
Correction (effective with 1.1.9):: The constraint originally given by $|abz|<|cq|$ has been corrected to read $|cq|<|abz|$ .
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

… ►

§17.6(iv) Differential Equations

… ►(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions

a\to q^{a}

b\to q^{b}

c\to q^{c}

, followed by

\lim_{q\to 1-}

. …

24: 13.29 Methods of Computation

… ►

§13.29(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. … ►For

M\left(a,b,z\right)

and

M_{\kappa,\mu}\left(z\right)

this means that in the sector

|\operatorname{ph}z|\leq\pi

we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). ►For

U\left(a,b,z\right)

and

W_{\kappa,\mu}\left(z\right)

we may integrate along outward rays from the origin in the sectors

\tfrac{1}{2}\pi<|\operatorname{ph}z|<\tfrac{3}{2}\pi

, with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). … ►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. …

25: Bibliography D

… ►

A. Debosscher (1998) Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

ⓘ

External Links:: ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

…

26: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►

§35.7(iii) Partial Differential Equations

… ►Subject to the conditions (a)–(c), the function

f(\mathbf{T})={{}_{2}F_{1}}\left(a,b;c;\mathbf{T}\right)

is the unique solution of each partial differential equation … ►Systems of partial differential equations for the

{{}_{0}F_{1}}

(defined in §35.8) and

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

functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …

27: 18.5 Explicit Representations

… ►

18.5.11_1

T_{n}\left(x\right)=\tfrac{1}{2}n\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}(n-\ell-1)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=2^{n-1}x^{n}{{}% _{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-n};\frac{1}% {x^{2}}\right),

n\geq 1

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Source:: Erdélyi et al. (1953b, 10.11(22))
Referenced by:: §18.5(iii), Erratum (V1.2.0) Section 18.5
Permalink:: http://dlmf.nist.gov/18.5.E11_1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

►

18.5.11_2

T_{n}\left(x\right)={{}_{2}F_{1}}\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}% \right),

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.5.E11_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

►

18.5.11_3

U_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{% \ell}(n-\ell)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=\left(2x\right)^{n}{{}_{2}F_{% 1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-n};\frac{1}{x^{2}}% \right),

ⓘ

Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Source:: Erdélyi et al. (1953b, 10.11(23))
Permalink:: http://dlmf.nist.gov/18.5.E11_3
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

►

18.5.11_4

U_{n}\left(x\right)=\left(n+1\right){{}_{2}F_{1}}\left({-n,n+2\atop\frac{3}{2}% };\frac{1-x}{2}\right).

ⓘ

Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.5(iii), Erratum (V1.2.0) Section 18.5
Permalink:: http://dlmf.nist.gov/18.5.E11_4
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

…

28: 10.16 Relations to Other Functions

… ►Principal values on each side of these equations correspond. ►

Confluent Hypergeometric Functions

… ►For the functions

M

and

U

see §13.2(i). …For the functions

M_{0,\nu}

and

W_{0,\nu}

see §13.14(i). … ►

Generalized Hypergeometric Functions

…

29: 18.35 Pollaczek Polynomials

… ►

18.35.4_5

P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=\frac{{\left(2\lambda\right)_{n}}% }{n!}\,{\mathrm{e}}^{\mathrm{i}n\theta}\*F\left({-n,\lambda+\mathrm{i}\tau_{a,% b}(\theta)\atop 2\lambda};1-{\mathrm{e}}^{-2\mathrm{i}\theta}\right).

ⓘ

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), $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$ : Pollaczek polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer and $\tau_{a,b}(\theta)$
Source:: Erdélyi et al. (1953b, 10.21(10))
Proof sketch:: Apply (15.8.7) to (18.35.4).
Referenced by:: (18.35.9), §18.35(i), Erratum (V1.2.0) Section 18.35
Permalink:: http://dlmf.nist.gov/18.35.E4_5
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.35(i), §18.35 and Ch.18

… ►

18.35.6_6

w^{(\lambda)}(\cos\theta;a,b,c)=\frac{{\mathrm{e}}^{(2\theta-\pi)\tau_{a,b}(% \theta)}\left(2\sin\theta\right)^{2\lambda-1}{\left|\Gamma\left(c+\lambda+% \mathrm{i}\tau_{a,b}(\theta)\right)\right|}^{2}}{\pi{\left|F\left({1-\lambda+% \mathrm{i}\tau_{a,b}(\theta),c\atop c+\lambda+\mathrm{i}\tau_{a,b}(\theta)};{% \mathrm{e}}^{2\mathrm{i}\theta}\right)\right|}^{2}},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $w(x)$ : weight function and $\tau_{a,b}(\theta)$
Source:: Pollaczek (1950, (2))
Referenced by:: §18.35(ii), Erratum (V1.2.0) Section 18.35
Permalink:: http://dlmf.nist.gov/18.35.E6_6
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.35(ii), §18.35 and Ch.18

…

30: 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

… ►The next expansion is given in Nørlund (1955, equation (1.21)): ►

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).