§35.6 Confluent Hypergeometric Functions of Matrix Argument

Keywords:: confluent hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.6

§35.6(i) Definitions
§35.6(ii) Properties
§35.6(iii) Relations to Bessel Functions of Matrix Argument
§35.6(iv) Asymptotic Approximations

§35.6(i) Definitions

Notes:: See Koecher (1954), Muirhead (1982, pp. 472–473). See also Herz (1955), Muirhead (1978).
Keywords:: confluent hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.6.i

35.6.1 $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a\atop b};\mathbf{T}\right)=\sum _{{k=0}}^{\infty}\frac{1}{k!}\sum _{{|\kappa|=k}}\frac{\left[a\right]_{{\kappa}}}{\left[b\right]_{{\kappa}}}\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $!$ : $n!$ : factorial, $\left[a\right]_{{\kappa}}$ : partitional shifted factorial, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $a$ : complex variable, $b$ : complex variable and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/35.6.E1
Encodings:: TeX, pMML, png

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:: $\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)$ : confluent hypergeometric function of matrix argument (second kind)
Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : 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

Keywords:: confluent hypergeometric functions

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E3
Encodings:: TeX, pMML, png

§35.6(ii) Properties

Notes:: See Herz (1955), Muirhead (1982, pp. 264–266, 472–473). For (35.6.6) apply (35.2.3) and (35.6.5). See also Shimura (1982).
Keywords:: confluent hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.6.ii

35.6.4 $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b-a\right)}\int\limits _{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{{a-\frac{1}{2}(m+1)}}|\mathbf{I}-\mathbf{X}|^{{b-a-\frac{1}{2}(m+1)}}d\mathbf{X},$ $\realpart{(a)},\realpart{(b-a)}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ : multivariate beta function, $\realpart{}$ : real part, $a$ : complex variable, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E4
Encodings:: TeX, pMML, png

35.6.5 $\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{{b-\frac{1}{2}(m+1)}}\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a\atop b};\mathbf{S}\mathbf{X}\right)d\mathbf{X}=\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b\right)|\mathbf{I}-\mathbf{S}\mathbf{T}^{{-1}}|^{{-a}}|\mathbf{T}|^{{-b}},$ $\mathbf{T}>\mathbf{S}$ , $\realpart{(b)}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part, $a$ : complex variable, ${\boldsymbol{\Omega}}$ : space of matrices, $b$ : complex variable and $m$ : positive integer
Referenced by:: §35.6(ii)
Permalink:: http://dlmf.nist.gov/35.6.E5
Encodings:: TeX, pMML, png

35.6.6 $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(b_{1},b_{2}\right)|\mathbf{T}|^{{b_{1}+b_{2}-\frac{1}{2}(m+1)}}\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}\right)=\int _{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}}|\mathbf{X}|^{{b_{1}-\frac{1}{2}(m+1)}}\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a_{1}\atop b_{1}};\mathbf{X}\right){|\mathbf{T}-\mathbf{X}|}^{{b_{2}-\frac{1}{2}(m+1)}}\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)d\mathbf{X},$ $\realpart{(b_{1})},\realpart{(b_{2})}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ : multivariate beta function, $\realpart{}$ : real part, $a$ : complex variable, $b$ : complex variable and $m$ : positive integer
Referenced by:: §35.6(ii)
Permalink:: http://dlmf.nist.gov/35.6.E6
Encodings:: TeX, pMML, png

35.6.7 $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({a\atop b};\mathbf{T}\right)=\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{T}\right)\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({b-a\atop b};-\mathbf{T}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $a$ : complex variable and $b$ : complex variable
Permalink:: http://dlmf.nist.gov/35.6.E7
Encodings:: TeX, pMML, png

35.6.8 $\int _{{\boldsymbol{\Omega}}}|\mathbf{T}|^{{c-\frac{1}{2}(m+1)}}\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)d\mathbf{T}=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a-c\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c-b+\frac{1}{2}(m+1)\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a-b+\frac{1}{2}(m+1)\right)},$ $\realpart{(a)}>\realpart{(c)}+\frac{1}{2}(m-1)>m-1$ , $\realpart{(c-b)}>-1$ .

Symbols:: $\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)$ : confluent hypergeometric function of matrix argument (second kind), $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part, $a$ : complex variable, ${\boldsymbol{\Omega}}$ : space of matrices, $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E8
Encodings:: TeX, pMML, png

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

Notes:: See Herz (1955), Muirhead (1978).
Keywords:: Bessel functions, confluent hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.6.iii

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

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E9
Encodings:: TeX, pMML, png

35.6.10 $\lim _{{a\to\infty}}\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)\mathop{\Psi\/}\nolimits\!\left(a+\nu;\nu+\tfrac{1}{2}(m+1);a^{{-1}}\mathbf{T}\right)=\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right).$

Symbols:: $\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (second kind), $\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)$ : confluent hypergeometric function of matrix argument (second kind), $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $a$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E10
Encodings:: TeX, pMML, png

§35.6(iv) Asymptotic Approximations

Keywords:: confluent hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.6.iv

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