35 Functions of Matrix Argument Properties 35.4 Partitions and Zonal Polynomials 35.6 Confluent Hypergeometric Functions of Matrix Argument

§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 : 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, : : factorial, $\left[a\right]_{{\kappa}}$ : partitional shifted factorial, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, : nonnegative integer, : positive integer and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Permalink:: http://dlmf.nist.gov/35.5.E2
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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), : differential of , $\in$ : element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, ${\boldsymbol{\Omega}}$ : space of matrices and : 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), : differential of , $\in$ : element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\realpart{}$ : real part, ${\boldsymbol{\Omega}}$ : space of matrices, : positive integer and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Permalink:: http://dlmf.nist.gov/35.5.E4
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35.5.5 $\int\limits_{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}}\mathop{A_{{\nu_{1}}}\/}% \nolimits\!\left(\mathbf{S}_{1}\mathbf{X}\right)|\mathbf{X}|^{{\nu_{1}}}\*% \mathop{A_{{\nu_{2}}}\/}\nolimits\!\left(\mathbf{S}_{2}(\mathbf{T}-\mathbf{X})% \right)|\mathbf{T}-\mathbf{X}|^{{\nu_{2}}}d\mathbf{X}=|\mathbf{T}|^{{\nu_{1}+% \nu_{2}+\frac{1}{2}(m+1)}}\mathop{A_{{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}}\/}% \nolimits\!\left((\mathbf{S}_{1}+\mathbf{S}_{2})\mathbf{T}\right),$ $\nu_{j}\in\Complex$ , $\realpart{(\nu_{j})}>-1$ ,

; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$ ; $\mathbf{T}\in{\boldsymbol{\Omega}}$ .

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $\Complex$ : complex plane (excluding infinity), : differential of , $\in$ : element of, $\int$ : integral, $\realpart{}$ : real part, ${\boldsymbol{\Omega}}$ : space of matrices, : nonnegative integer, : 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), : differential of , $\in$ : element of, $\int$ : integral, $\realpart{}$ : real part, ${\boldsymbol{\Omega}}$ : space of matrices and : 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), : differential of , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathbf{O}(m)$ : space of orthogonal matrices and : 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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 35.4 Partitions and Zonal Polynomials 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

Contents

§35.5(i) Definitions

§35.5(ii) Properties

§35.5(iii) Asymptotic Approximations