# 3F2 functions of matrix argument

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##### 1: 35.1 Special Notation
(For other notation see Notation for the Special Functions.) …
 $a,b$ complex variables. …
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) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ 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 $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. 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)).
##### 2: 35.5 Bessel Functions of Matrix Argument
###### §35.5(iii) Asymptotic Approximations
For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
##### 6: 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).
##### 7: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $k$ nonnegative integer, except in §9.9(iii). … derivatives with respect to argument.
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)).
##### 8: 9.12 Scorer Functions
###### §9.12 Scorer Functions
If $\zeta=\tfrac{2}{3}z^{3/2}$ or $\tfrac{2}{3}x^{3/2}$, and $K_{1/3}$ is the modified Bessel function10.25(ii)), then … where the integration contour separates the poles of $\Gamma\left(\tfrac{1}{3}+\tfrac{1}{3}t\right)$ from those of $\Gamma\left(-t\right)$. …
##### 9: 20.2 Definitions and Periodic Properties
###### §20.2(i) Fourier Series
Corresponding expansions for $\theta_{j}'\left(z\middle|\tau\right)$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. … For fixed $z$, each of $\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}$, $\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}$, $\theta_{3}\left(z\middle|\tau\right)$, and $\theta_{4}\left(z\middle|\tau\right)$ is an analytic function of $\tau$ for $\Im\tau>0$, with a natural boundary $\Im\tau=0$, and correspondingly, an analytic function of $q$ for $\left|q\right|<1$ with a natural boundary $\left|q\right|=1$. …
###### §20.2(iii) Translation of the Argument by Half-Periods
For $m,n\in\mathbb{Z}$, the $z$-zeros of $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, are $(m+n\tau)\pi$, $(m+\tfrac{1}{2}+n\tau)\pi$, $(m+\tfrac{1}{2}+(n+\tfrac{1}{2})\tau)\pi$, $(m+(n+\tfrac{1}{2})\tau)\pi$ respectively.