§35.5 Bessel Functions of Matrix Argument

35.5.1		$$A_{\nu }\left(\mathbf{0}\right)=\frac{1}{{\mathrm{\Gamma }}_m\left(\nu +\frac{1}{2}(m+1)\right)},$$
$\nu \in ℂ$.
ⓘ Symbols: $A_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $m$: positive integer and $\mathbf{0}$: $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.5.E1 Encodings: TeX, pMML, png See also: Annotations for §35.5(i), §35.5 and Ch.35

35.5.2		$$A_{\nu }\left(\mathbf{T}\right)=A_{\nu }\left(\mathbf{0}\right)\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\sum _{|\kappa |=k}\frac{1}{{\left[\nu +\frac{1}{2}(m+1)\right]}_{\kappa }}{Z}_{\kappa }\left(\mathbf{T}\right),$$
$\nu \in ℂ$, $\mathbf{T}\in \mathbf{𝒮}$.
ⓘ Symbols: $A_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $\mathbf{𝒮}$: space of real symmetric $m\times m$ matrices, $\mathbf{T}$: real symmetric $m\times m$ matrix, $k$: nonnegative integer, $m$: positive integer, $\mathbf{0}$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.5.E2 Encodings: TeX, pMML, png See also: Annotations for §35.5(i), §35.5 and Ch.35

35.5.3		$$B_{\nu }\left(\mathbf{T}\right)={\int }_{\mathbf{\Omega }}etr\left(-(\mathbf{T}\mathbf{X}+{\mathbf{X}}^{-1})\right){\left|\mathbf{X}\right|}^{\nu -\frac{1}{2}(m+1)}\mathrm{d}\mathbf{X},$$
$\nu \in ℂ$, $\mathbf{T}\in \mathbf{\Omega }$.
ⓘ Defines: $B_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (second kind) Symbols: $ℂ$: complex plane, $\in$: element of, $etr\left(\mathbf{X}\right)$: exponential of trace, $\int$: integral, $\mathbf{T}$: 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 and $m$: positive integer Referenced by: §35.9 Permalink: http://dlmf.nist.gov/35.5.E3 Encodings: TeX, pMML, png See also: Annotations for §35.5(i), §35.5 and Ch.35

35.5.4		$${\int }_{\mathbf{\Omega }}etr\left(-\mathbf{T}\mathbf{X}\right){\left|\mathbf{X}\right|}^{\nu }{A}_{\nu }\left(\mathbf{S}\mathbf{X}\right)\mathrm{d}\mathbf{X}=etr\left(-\mathbf{S}{\mathbf{T}}^{-1}\right){\left|\mathbf{T}\right|}^{-\nu -\frac{1}{2}(m+1)},$$
$\mathbf{S}\in \mathbf{𝒮}$, $\mathbf{T}\in \mathbf{\Omega }$; $\mathrm{\Re }\left(\nu \right)>-1$.
ⓘ Symbols: $A_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (first kind), $\in$: element of, $etr\left(\mathbf{X}\right)$: exponential of trace, $\int$: integral, $\mathrm{\Re }$: real part, $\mathbf{𝒮}$: space of real symmetric $m\times m$ matrices, $\mathbf{S}$: 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 and $m$: positive integer Permalink: http://dlmf.nist.gov/35.5.E4 Encodings: TeX, pMML, png See also: Annotations for §35.5(ii), §35.5 and Ch.35

35.5.5
${\nu }_j\in ℂ$, $\mathrm{\Re }\left({\nu }_j\right)>-1$, $j=1,2$; ${\mathbf{S}}_1,{\mathbf{S}}_2\in \mathbf{𝒮}$; $\mathbf{T}\in \mathbf{\Omega }$.
ⓘ Symbols: $A_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $\mathbf{𝒮}$: space of real symmetric $m\times m$ matrices, $\mathbf{S}$: 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, : less-than for matrices, $j$: nonnegative integer, $m$: positive integer and $\mathbf{0}$: $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.5.E5 Encodings: TeX, pMML, png See also: Annotations for §35.5(ii), §35.5 and Ch.35

35.5.6		$$B_{\nu }\left(\mathbf{T}\right)={\left|\mathbf{T}\right|}^{-\nu }{B}_{-\nu }\left(\mathbf{T}\right),$$
$\nu \in ℂ$, $\mathbf{T}\in \mathbf{\Omega }$.
ⓘ Symbols: $B_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (second kind), $ℂ$: complex plane, $\in$: element of, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$ and $\mathbf{\Omega }$: space of positive-definite real symmetric $m\times m$ matrices Permalink: http://dlmf.nist.gov/35.5.E6 Encodings: TeX, pMML, png See also: Annotations for §35.5(ii), §35.5 and Ch.35

35.5.7		$${\int }_{\mathbf{\Omega }}{A}_{{\nu }_1}\left(\mathbf{T}\mathbf{X}\right)B_{-{\nu }_2}\left(\mathbf{S}\mathbf{X}\right){\left|\mathbf{X}\right|}^{{\nu }_1}\mathrm{d}\mathbf{X}=\frac{1}{A_{{\nu }_1+{\nu }_2}\left(\mathbf{0}\right)}{\left|\mathbf{S}\right|}^{{\nu }_2}{\left|\mathbf{T}+\mathbf{S}\right|}^{-({\nu }_1+{\nu }_2+\frac{1}{2}(m+1))},$$
$\mathrm{\Re }\left({\nu }_1+{\nu }_2\right)>-1$; $\mathbf{S},\mathbf{T}\in \mathbf{\Omega }$.
ⓘ Symbols: $A_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (first kind), $B_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (second kind), $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $\mathbf{S}$: 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, $m$: positive integer and $\mathbf{0}$: $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.5.E7 Encodings: TeX, pMML, png See also: Annotations for §35.5(ii), §35.5 and Ch.35

35.5.8		$${\int }_{\mathbf{O}(m)}etr\left(\mathbf{S}\mathbf{H}\right)\mathrm{d}\mathbf{H}=\frac{A_{-1/2}\left(-\frac{1}{4}\mathbf{S}{\mathbf{S}}^{\mathrm{T}}\right)}{A_{-1/2}\left(\mathbf{0}\right)},$$
$\mathbf{S}$ arbitrary.
ⓘ Symbols: $A_{\nu }\left(\mathbf{T}\right)$: Bessel function of matrix argument (first kind), $etr\left(\mathbf{X}\right)$: exponential of trace, $\int$: integral, $\mathbf{S}$: real symmetric $m\times m$ matrix, $\mathbf{O}(m)$: space of $m\times m$ orthogonal matrices, $\mathbf{H}$: orthogonal $m\times m$ matrix, $\mathrm{d}\mathbf{H}$: normalized Haar measure on $\mathbf{O}(m)$, $m$: positive integer and $\mathbf{0}$: $m\times m$ matrix of zeros Referenced by: §35.10, §35.9 Permalink: http://dlmf.nist.gov/35.5.E8 Encodings: TeX, pMML, png See also: Annotations for §35.5(ii), §35.5 and Ch.35

For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).