for 3F2 hypergeometric functions of matrix argument

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1: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

►The hypergeometric function

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

is defined by the Gauss series … … ►On the circle of convergence,

|z|=1

, the Gauss series: … ►

§15.2(ii) Analytic Properties

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2: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(iii) ${{}_{3}F_{2}}$ Case

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Kummer Transformation

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Pfaff–Saalschütz Formula

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Thomae Transformation

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3: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(i) Definition

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Jacobi Form

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Confluent Form

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Integral Representation

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4: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

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§35.6(i) Definitions

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Laguerre Form

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§35.6(ii) Properties

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§35.6(iii) Relations to Bessel Functions of Matrix Argument

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5: 34.2 Definition: $\mathit{3j}$ Symbol

§34.2 Definition: $\mathit{3j}$ Symbol

►The quantities

j_{1},j_{2},j_{3}

in the

\mathit{3j}

symbol are called angular momenta. …The corresponding projective quantum numbers

m_{1},m_{2},m_{3}

are given by … ►where

{{}_{3}F_{2}}

is defined as in §16.2. ►For alternative expressions for the

\mathit{3j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{3}F_{2}}

of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

6: 17.1 Special Notation

§17.1 Special Notation

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$k, j, m, n, r, s$	nonnegative integers.
…

►The main functions treated in this chapter are the basic hypergeometric (or

q

-hypergeometric) function

{{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, the bilateral basic hypergeometric (or bilateral

q

-hypergeometric) function

{{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, and the

q

-analogs of the Appell functions

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)

, and

\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)

. … ►Fine (1988) uses

F(a,b;t:q)

for a particular specialization of a

{{}_{2}\phi_{1}}

function.

7: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

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Polynomials

… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►

§16.2(v) Behavior with Respect to Parameters

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8: 35.1 Special Notation

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

$a, b$	complex variables.
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively

\Gamma_{m}\left(a\right)

and

\mathrm{B}_{m}\left(a,b\right)

, and the special functions of matrix argument: Bessel (of the first kind)

A_{\nu}\left(\mathbf{T}\right)

and (of the second kind)

B_{\nu}\left(\mathbf{T}\right)

; confluent hypergeometric (of the first kind)

{{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)

\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)

and (of the second kind)

\Psi\left(a;b;\mathbf{T}\right)

; Gaussian hypergeometric

{{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)

\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)

; generalized hypergeometric

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

\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)

. ►An alternative notation for the multivariate gamma function is

\Pi_{m}(a)=\Gamma_{m}\left(a+\tfrac{1}{2}(m+1)\right)

(Herz (1955, p. 480)). Related notations for the Bessel functions are

\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)

(Faraut and Korányi (1994, pp. 320–329)),

K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)

(Terras (1988, pp. 49–64)), and

\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)

(Faraut and Korányi (1994, pp. 357–358)).

9: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

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§35.5(i) Definitions

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§35.5(ii) Properties

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§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

10: 19.16 Definitions

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§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 …The

R

-function is often used to make a unified statement of a property of several elliptic integrals. … ► ►

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

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