§35.5 Bessel Functions of Matrix Argument

Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/35.5

§35.5(i) Definitions
§35.5(ii) Properties
§35.5(iii) Asymptotic Approximations

§35.5(i) Definitions

Notes:: See Bochner (1952), Herz (1955), Gross and Kunze (1976).
Defines:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind)
Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/35.5.i

35.5.1 $\mathop{A_{{\nu}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\frac{1}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(\nu+\frac{1}{2}(m+1)\right)},$ $\nu\in\Complex$ .

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.5.E1
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35.5.2 $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)=\mathop{A_{{\nu}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)\sum _{{k=0}}^{\infty}\frac{(-1)^{k}}{k!}\sum _{{|\kappa|=k}}\frac{1}{\left[\nu+\frac{1}{2}(m+1)\right]_{{\kappa}}}\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right),$ $\nu\in\Complex$ , $\mathbf{T}\in\boldsymbol{\mathcal{S}}$ .

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $!$ : $n!$ : factorial, $\left[a\right]_{{\kappa}}$ : partitional shifted factorial, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $k$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Permalink:: http://dlmf.nist.gov/35.5.E2
Encodings:: TeX, pMML, png

35.5.3 $\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)=\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-(\mathbf{T}\mathbf{X}+\mathbf{X}^{{-1}})\right)|\mathbf{X}|^{{\nu-\frac{1}{2}(m+1)}}d\mathbf{X},$ $\nu\in\Complex$ , $\mathbf{T}\in{\boldsymbol{\Omega}}$ .

Defines:: $\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (second kind)
Symbols:: $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, ${\boldsymbol{\Omega}}$ : space of matrices and $m$ : positive integer
Referenced by:: §35.9
Permalink:: http://dlmf.nist.gov/35.5.E3
Encodings:: TeX, pMML, png

§35.5(ii) Properties

Notes:: See Herz (1955). See also Terras (1988, pp. 49–63), Butler and Wood (2003).
Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/35.5.ii

35.5.4 $\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{\nu}\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{S}\mathbf{X}\right)d\mathbf{X}=\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{S}\mathbf{T}^{{-1}}\right)|\mathbf{T}|^{{-\nu-\frac{1}{2}(m+1)}},$ $\mathbf{S}\in\boldsymbol{\mathcal{S}}$ , $\mathbf{T}\in{\boldsymbol{\Omega}}$ ; $\realpart{(\nu)}>-1$ .

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $dx$ : differential of $x$ , $\in$ : element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\realpart{}$ : real part, ${\boldsymbol{\Omega}}$ : space of matrices, $m$ : positive integer and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Permalink:: http://dlmf.nist.gov/35.5.E4
Encodings:: TeX, pMML, png

35.5.5 $\int\limits _{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}}\mathop{A_{{\nu _{1}}}\/}\nolimits\!\left(\mathbf{S}_{1}\mathbf{X}\right)|\mathbf{X}|^{{\nu _{1}}}\*\mathop{A_{{\nu _{2}}}\/}\nolimits\!\left(\mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)|\mathbf{T}-\mathbf{X}|^{{\nu _{2}}}d\mathbf{X}=|\mathbf{T}|^{{\nu _{1}+\nu _{2}+\frac{1}{2}(m+1)}}\mathop{A_{{\nu _{1}+\nu _{2}+\frac{1}{2}(m+1)}}\/}\nolimits\!\left((\mathbf{S}_{1}+\mathbf{S}_{2})\mathbf{T}\right),$ $\nu _{j}\in\Complex$ , $\realpart{(\nu _{j})}>-1$ , $j=1,2$ ; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$ ; $\mathbf{T}\in{\boldsymbol{\Omega}}$ .

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\realpart{}$ : real part, ${\boldsymbol{\Omega}}$ : space of matrices, $j$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Permalink:: http://dlmf.nist.gov/35.5.E5
Encodings:: TeX, pMML, png

35.5.6 $\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)=|\mathbf{T}|^{{-\nu}}\mathop{B_{{-\nu}}\/}\nolimits\!\left(\mathbf{T}\right),$ $\nu\in\Complex$ , $\mathbf{T}\in{\boldsymbol{\Omega}}$ .

Symbols:: $\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (second kind), $\Complex$ : complex plane (excluding infinity), $\in$ : element of and ${\boldsymbol{\Omega}}$ : space of matrices
Permalink:: http://dlmf.nist.gov/35.5.E6
Encodings:: TeX, pMML, png

35.5.7 $\int _{{\boldsymbol{\Omega}}}\mathop{A_{{\nu _{1}}}\/}\nolimits\!\left(\mathbf{T}\mathbf{X}\right)\mathop{B_{{-\nu _{2}}}\/}\nolimits\!\left(\mathbf{S}\mathbf{X}\right)|\mathbf{X}|^{{\nu _{1}}}d\mathbf{X}=\frac{1}{\mathop{A_{{\nu _{1}+\nu _{2}}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)}|\mathbf{S}|^{{\nu _{2}}}|\mathbf{T}+\mathbf{S}|^{{-(\nu _{1}+\nu _{2}+\frac{1}{2}(m+1))}},$ $\realpart{(\nu _{1}+\nu _{2})}>-1$ ; $\mathbf{S},\mathbf{T}\in{\boldsymbol{\Omega}}$ .

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $\mathop{B_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (second kind), $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\realpart{}$ : real part, ${\boldsymbol{\Omega}}$ : space of matrices and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.5.E7
Encodings:: TeX, pMML, png

35.5.8 $\int _{{\mathbf{O}(m)}}\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{S}\mathbf{H}\right)d\mathbf{H}=\frac{\mathop{A_{{-1/2}}\/}\nolimits\!\left(-\frac{1}{4}\mathbf{S}\mathbf{S}^{\mathrm{T}}\right)}{\mathop{A_{{-1/2}}\/}\nolimits\!\left(\boldsymbol{{0}}\right)},$ $\mathbf{S}$ arbitrary.

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathbf{O}(m)$ : space of orthogonal matrices and $m$ : positive integer
Referenced by:: §35.10, §35.9
Permalink:: http://dlmf.nist.gov/35.5.E8
Encodings:: TeX, pMML, png

§35.5(iii) Asymptotic Approximations

Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/35.5.iii

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

§35.5 Bessel Functions of Matrix Argument

Contents

§35.5(i) Definitions

§35.5(ii) Properties

§35.5(iii) Asymptotic Approximations