§35.6 Confluent Hypergeometric Functions of Matrix Argument

35.6.1		$${}_{1}F_1(\genfrac{}{}{0pt}{}{a}{b};\mathbf{T})=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }}{{\left[b\right]}_{\kappa }}{Z}_{\kappa }\left(\mathbf{T}\right).$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.6.E1 Encodings: TeX, pMML, png See also: Annotations for §35.6(i), §35.6 and Ch.35

35.6.2		$$\mathrm{\Psi }(a;b;\mathbf{T})=\frac{1}{{\mathrm{\Gamma }}_m\left(a\right)}{\int }_{\mathbf{\Omega }}etr\left(-\mathbf{T}\mathbf{X}\right){\left|\mathbf{X}\right|}^{a-\frac{1}{2}(m+1)}{\left|\mathbf{I}+\mathbf{X}\right|}^{b-a-\frac{1}{2}(m+1)}\mathrm{d}\mathbf{X},$$
$\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$, $\mathbf{T}\in \mathbf{\Omega }$.
ⓘ Defines: $\mathrm{\Psi }(a;b;\mathbf{T})$: confluent hypergeometric function of matrix argument (second kind) Symbols: $\in$: element of, $etr\left(\mathbf{X}\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{X}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$, $\mathrm{d}$: differential of matrix, $\mathbf{\Omega }$: space of positive-definite real symmetric $m\times m$ 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 Ch.35

35.6.4
$\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b-a\right)>\frac{1}{2}(m-1)$.
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $etr\left(\mathbf{X}\right)$: exponential of trace, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{X}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$, $\mathrm{d}$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $\mathbf{0}$: $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.6.E4 Encodings: TeX, pMML, png See also: Annotations for §35.6(ii), §35.6 and Ch.35

35.6.5		$${\int }_{\mathbf{\Omega }}etr\left(-\mathbf{T}\mathbf{X}\right){\left|\mathbf{X}\right|}^{b-\frac{1}{2}(m+1)}{}_{1}F_1(\genfrac{}{}{0pt}{}{a}{b};\mathbf{S}\mathbf{X})\mathrm{d}\mathbf{X}={\mathrm{\Gamma }}_m\left(b\right){\left|\mathbf{I}-\mathbf{S}{\mathbf{T}}^{-1}\right|}^{-a}{\left|\mathbf{T}\right|}^{-b},$$
$\mathbf{T}\mathit{>}\mathbf{S}$, $\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $etr\left(\mathbf{X}\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{S}$: real symmetric $m\times m$ matrix, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{X}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$, $\mathrm{d}$: differential of matrix, $\mathbf{\Omega }$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $\mathit{>}$: greater-than for matrices and $m$: positive integer Referenced by: §35.6(ii) Permalink: http://dlmf.nist.gov/35.6.E5 Encodings: TeX, pMML, png See also: Annotations for §35.6(ii), §35.6 and Ch.35

35.6.6
$\mathrm{\Re }\left(b_1\right),\mathrm{\Re }\left(b_2\right)>\frac{1}{2}(m-1)$.
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{X}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$, $\mathrm{d}$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $\mathbf{0}$: $m\times m$ matrix of zeros Referenced by: §35.6(ii) Permalink: http://dlmf.nist.gov/35.6.E6 Encodings: TeX, pMML, png See also: Annotations for §35.6(ii), §35.6 and Ch.35

35.6.7		$${}_{1}F_1(\genfrac{}{}{0pt}{}{a}{b};\mathbf{T})=etr\left(\mathbf{T}\right){}_{1}F_1(\genfrac{}{}{0pt}{}{b-a}{b};-\mathbf{T}).$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $etr\left(\mathbf{X}\right)$: exponential of trace, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix and $b$: complex variable Permalink: http://dlmf.nist.gov/35.6.E7 Encodings: TeX, pMML, png See also: Annotations for §35.6(ii), §35.6 and Ch.35

35.6.8		$${\int }_{\mathbf{\Omega }}{\left|\mathbf{T}\right|}^{c-\frac{1}{2}(m+1)}\mathrm{\Psi }(a;b;\mathbf{T})\mathrm{d}\mathbf{T}=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(a-c\right){\mathrm{\Gamma }}_m\left(c-b+\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(a-b+\frac{1}{2}(m+1)\right)},$$
$\mathrm{\Re }\left(a\right)>\mathrm{\Re }\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\mathrm{\Re }\left(c-b\right)>-1$.
ⓘ Symbols: $\mathrm{\Psi }(a;b;\mathbf{T})$: confluent hypergeometric function of matrix argument (second kind), $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$, $\mathrm{d}$: differential of matrix, $\mathbf{\Omega }$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $c$: complex variable and $m$: positive integer Permalink: http://dlmf.nist.gov/35.6.E8 Encodings: TeX, pMML, png See also: Annotations for §35.6(ii), §35.6 and Ch.35

35.6.9		$$\underset{a\to \mathrm{\infty }}{lim}{}_{1}F_1(\genfrac{}{}{0pt}{}{a}{\nu +\frac{1}{2}(m+1)};-a^{-1}\mathbf{T})=\frac{A_{\nu }\left(\mathbf{T}\right)}{A_{\nu }\left(\mathbf{0}\right)}.$$
ⓘ Symbols: $A_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (first kind), ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix, $m$: positive integer and $\mathbf{0}$: $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.6.E9 Encodings: TeX, pMML, png See also: Annotations for §35.6(iii), §35.6 and Ch.35

35.6.10		$$\underset{a\to \mathrm{\infty }}{lim}{\mathrm{\Gamma }}_m\left(a\right)\mathrm{\Psi }(a+\nu ;\nu +\frac{1}{2}(m+1);a^{-1}\mathbf{T})=B_{\nu }\left(\mathbf{T}\right).$$
ⓘ Symbols: $B_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (second kind), $\mathrm{\Psi }(a;b;\mathbf{T})$: confluent hypergeometric function of matrix argument (second kind), ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix and $m$: positive integer Permalink: http://dlmf.nist.gov/35.6.E10 Encodings: TeX, pMML, png See also: Annotations for §35.6(iii), §35.6 and Ch.35

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