# §35.6 Confluent Hypergeometric Functions of Matrix Argument

## §35.6(i) Definitions

 35.6.1 $\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left({a\atop b};\mathbf{T}\right)=\sum_{k=% 0}^{\infty}\frac{1}{k!}\sum_{|\kappa|=k}\frac{\left[a\right]_{\kappa}}{\left[b% \right]_{\kappa}}\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{T}\right).$
 35.6.2 $\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)=\frac{1}{\mathop{\Gamma_% {m}\/}\nolimits\!\left(a\right)}\int_{\boldsymbol{\Omega}}\mathop{\mathrm{etr}% \/}\nolimits\!\left(-\mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{a-\frac{1}{2}(m+% 1)}\*{|\mathbf{I}+\mathbf{X}|}^{b-a-\frac{1}{2}(m+1)}d\mathbf{X},$ $\realpart{(a)}>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$. Defines: $\mathop{\Psi\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$: confluent hypergeometric function of matrix argument (second kind) Symbols: $d\NVar{x}$: differential of $x$, $\in$: element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\NVar{\mathbf{X}}\right)$: exponential of trace, $\int$: integral, $\mathop{\Gamma_{\NVar{m}}\/}\nolimits\!\left(\NVar{a}\right)$: multivariate gamma function, $\realpart{}$: 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

### Laguerre Form

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

## §35.6(ii) Properties

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

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

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

## §35.6(iv) Asymptotic Approximations

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