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
โบThe hypergeometric function $F\left(a,b;c;z\right)$ is defined by the Gauss series … … โบ
§15.2(ii) Analytic Properties
โบThe same properties hold for $F\left(a,b;c;z\right)$, except that as a function of $c$, $F\left(a,b;c;z\right)$ in general has poles at $c=0,-1,-2,\dots$. …
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5: 34.2 Definition: $\mathit{3j}$ Symbol
§34.2 Definition: $\mathit{3j}$ Symbol
โบThe 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 … โบwhere ${{}_{3}F_{2}}$ is defined as in §16.2. โบFor 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. …
โบThe 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 $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)$. … โบFine (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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Polynomials
โบNote also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … โบ
8: 35.1 Special Notation
โบ(For other notation see Notation for the Special Functions.) … โบ โบโบ
 $a,b$ complex variables. …
โบThe 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) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ or $\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\Psi\left(a;b;\mathbf{T}\right)$; Gaussian hypergeometric ${{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$; generalized hypergeometric ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. โบAn 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
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§35.5(iii) Asymptotic Approximations
โบFor 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) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$
โบAll 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. … โบ โบ