generalized%20hypergeometric%20function%200F2

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11: 19.16 Definitions

… ►

§19.16(ii) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function … ►For generalizations and further information, especially representation of the

R

-function as a Dirichlet average, see Carlson (1977b). ►

§19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

…

12: 19.2 Definitions

… ►

§19.2(i) General Elliptic Integrals

►Let

s^{2}(t)

be a cubic or quartic polynomial in

t

with simple zeros, and let

r(s,t)

be a rational function of

s

and

t

containing at least one odd power of

s

. … ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

… ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When

x

and

y

are positive,

R_{C}\left(x,y\right)

is an inverse circular function if

x<y

and an inverse hyperbolic function (or logarithm) if

x>y

: …

13: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

►

§14.19(i) Introduction

… ►

§14.19(ii) Hypergeometric Representations

►With

\mathbf{F}

as in §14.3 and

\xi>0

, … ►

§14.19(v) Whipple’s Formula for Toroidal Functions

…

14: 9.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►

$k$	nonnegative integer, except in §9.9(iii).
…

►The main functions treated in this chapter are the Airy functions

\operatorname{Ai}\left(z\right)

and

\operatorname{Bi}\left(z\right)

, and the Scorer functions

\operatorname{Gi}(z)

and

\operatorname{Hi}(z)

(also known as inhomogeneous Airy functions). ►Other notations that have been used are as follows:

\operatorname{Ai}\left(-x\right)

and

\operatorname{Bi}\left(-x\right)

for

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

(Jeffreys (1928), later changed to

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

);

U(x)=\sqrt{\pi}\operatorname{Bi}\left(x\right)

V(x)=\sqrt{\pi}\operatorname{Ai}\left(x\right)

(Fock (1945));

A(x)=3^{-\ifrac{1}{3}}\pi\operatorname{Ai}\left(-3^{-\ifrac{1}{3}}x\right)

(Szegő (1967, §1.81));

e_{0}(x)=\pi\operatorname{Hi}(-x)

\widetilde{e}_{0}(x)=-\pi\operatorname{Gi}(-x)

(Tumarkin (1959)).

15: 16.13 Appell Functions

§16.13 Appell Functions

►The following four functions of two real or complex variables

x

and

y

cannot be expressed as a product of two

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

functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►

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

… ►

16.13.4

{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

ⓘ

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

… ► …

16: 23.15 Definitions

§23.15 Definitions

►

§23.15(i) General Modular Functions

… ►

Elliptic Modular Function

… ►

Dedekind’s Eta Function (or Dedekind Modular Function)

… ►

17: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

… ► … ►Lastly, for the range

1<x<\infty

P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

is a real-valued solution of (14.20.1); in terms of

Q^{\mu}_{-\frac{1}{2}\pm\mathrm{i}\tau}\left(x\right)

(which are complex-valued in general): … ►

§14.20(ii) Graphics

… ►

§14.20(vi) Generalized Mehler–Fock Transformation

…

18: 11.9 Lommel Functions

§11.9 Lommel Functions

… ►can be regarded as a generalization of (11.2.7). Provided that

\mu\pm\nu\neq-1,-3,-5,\dots

, (11.9.1) has the general solution … ► … ►

19: 20.2 Definitions and Periodic Properties

… ►

§20.2(i) Fourier Series

… ►

§20.2(ii) Periodicity and Quasi-Periodicity

… ►The theta functions are quasi-periodic on the lattice: … ►

§20.2(iii) Translation of the Argument by Half-Periods

… ►

§20.2(iv) $z$ -Zeros

…

20: 9.12 Scorer Functions

§9.12 Scorer Functions

… ►The general solution is given by …where … ► … ►

Functions and Derivatives

…