# §35.6 Confluent Hypergeometric Functions of Matrix Argument

## §35.6(i) Definitions

 35.6.1 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!% }\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}}{{\left[b\right]_{\kappa}}}Z% _{\kappa}\left(\mathbf{T}\right).$
 35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)|\mathbf{X}|% ^{a-\frac{1}{2}(m+1)}\*{|\mathbf{I}+\mathbf{X}|}^{b-a-\frac{1}{2}(m+1)}\mathrm% {d}\mathbf{X},$ $\Re(a)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$. ⓘ Defines: $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$: confluent hypergeometric function of matrix argument (second kind) Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\in$: element of, $\mathrm{etr}\left(\NVar{\mathbf{X}}\right)$: exponential of trace, $\int$: integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$: multivariate gamma function, $\Re$: real part, $a$: complex variable, ${\boldsymbol{\Omega}}$: space of matrices, $b$: complex variable and $m$: positive integer Permalink: http://dlmf.nist.gov/35.6.E2 Encodings: TeX, pMML, png See also: Annotations for 35.6(i), 35.6 and 35

### Laguerre Form

 35.6.3 $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+\nu+% \frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*{{}_% {1}F_{1}}\left({-\nu\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),$ $\Re(\gamma),\Re(\gamma+\nu)>-1$.

## §35.6(ii) Properties

 35.6.4 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\mathrm{etr}% \left(\mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{a-\frac{1}{2}(m+1)}|\mathbf{I}-% \mathbf{X}|^{b-a-\frac{1}{2}(m+1)}\mathrm{d}\mathbf{X},$ $\Re(a),\Re(b-a)>\frac{1}{2}(m-1)$.
 35.6.5 $\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)|% \mathbf{X}|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};\mathbf{S}% \mathbf{X}\right)\mathrm{d}\mathbf{X}=\Gamma_{m}\left(b\right)|\mathbf{I}-% \mathbf{S}\mathbf{T}^{-1}|^{-a}|\mathbf{T}|^{-b},$ $\mathbf{T}>\mathbf{S}$, $\Re(b)>\frac{1}{2}(m-1)$.
 35.6.6 $\mathrm{B}_{m}\left(b_{1},b_{2}\right)|\mathbf{T}|^{b_{1}+b_{2}-\frac{1}{2}(m+% 1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}\right)=\int_{% \boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}|\mathbf{X}|^{b_{1}-\frac{1}{2}(m+1)}{{% }_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right){|\mathbf{T}-\mathbf{X}|}% ^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{2}\atop b_{2}};\mathbf{T}-% \mathbf{X}\right)\mathrm{d}\mathbf{X},$ $\Re(b_{1}),\Re(b_{2})>\frac{1}{2}(m-1)$.
 35.6.7 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\mathrm{etr}\left(\mathbf{T}% \right){{}_{1}F_{1}}\left({b-a\atop b};-\mathbf{T}\right).$
 35.6.8 $\int_{\boldsymbol{\Omega}}|\mathbf{T}|^{c-\frac{1}{2}(m+1)}\Psi\left(a;b;% \mathbf{T}\right)\mathrm{d}\mathbf{T}=\frac{\Gamma_{m}\left(c\right)\Gamma_{m}% \left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(a% \right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},$ $\Re(a)>\Re(c)+\frac{1}{2}(m-1)>m-1$, $\Re(c-b)>-1$.

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

 35.6.9 $\lim_{a\to\infty}{{}_{1}F_{1}}\left({a\atop\nu+\frac{1}{2}(m+1)};-a^{-1}% \mathbf{T}\right)=\frac{A_{\nu}\left(\mathbf{T}\right)}{A_{\nu}\left(% \boldsymbol{{0}}\right)}.$
 35.6.10 $\lim_{a\to\infty}\Gamma_{m}\left(a\right)\Psi\left(a+\nu;\nu+\tfrac{1}{2}(m+1)% ;a^{-1}\mathbf{T}\right)=B_{\nu}\left(\mathbf{T}\right).$

## §35.6(iv) Asymptotic Approximations

For asymptotic approximations for confluent hypergeometric functions of matrix argument, see Herz (1955) and Butler and Wood (2002).