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

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Keywords:: Bessel functions, of matrix argument
Permalink:: http://dlmf.nist.gov/35.5
See also:: Annotations for Ch.35

Contents

§35.5(i) Definitions

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Defines:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind)
Keywords:: Bessel functions, definitions, of matrix argument
Notes:: See Bochner (1952), Herz (1955), Gross and Kunze (1976).
Permalink:: http://dlmf.nist.gov/35.5.i
See also:: Annotations for §35.5 and Ch.35

35.5.1		$A_{\nu}\left(\boldsymbol{{0}}\right)=\frac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(% m+1)\right)},$
$\nu\in\mathbb{C}$ .
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\mathbb{C}$ : complex plane, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ 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		$A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{\|\kappa\|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),$
$\nu\in\mathbb{C}$ , $\mathbf{T}\in\boldsymbol{\mathcal{S}}$ .
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\mathbb{C}$ : complex plane, $\in$ : element of, $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $k$ : nonnegative integer, $m$ : positive integer, $\boldsymbol{{0}}$ : $m\times m$ 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		$B_{\nu}\left(\mathbf{T}\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}% \left(-(\mathbf{T}\mathbf{X}+\mathbf{X}^{-1})\right)\left\|\mathbf{X}\right\|^{% \nu-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$
$\nu\in\mathbb{C}$ , $\mathbf{T}\in{\boldsymbol{\Omega}}$ .
ⓘ Defines: $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (second kind) Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : 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(ii) Properties

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Keywords:: Bessel functions, of matrix argument, properties
Notes:: See Herz (1955). See also Terras (1988, pp. 49–63), Butler and Wood (2003).
Permalink:: http://dlmf.nist.gov/35.5.ii
See also:: Annotations for §35.5 and Ch.35

35.5.4		$\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left\|\mathbf{X}\right\|^{\nu}A_{\nu}\left(\mathbf{S}\mathbf{X}\right)\,\mathrm% {d}{\mathbf{X}}=\operatorname{etr}\left(-\mathbf{S}\mathbf{T}^{-1}\right)\left% \|\mathbf{T}\right\|^{-\nu-\frac{1}{2}(m+1)},$
$\mathbf{S}\in\boldsymbol{\mathcal{S}}$ , $\mathbf{T}\in{\boldsymbol{\Omega}}$ ; $\Re\left(\nu\right)>-1$ .
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\in$ : element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Re$ : real part, $\boldsymbol{\mathcal{S}}$ : 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, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : 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		$\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}A_{\nu_{1}}\left(\mathbf{S% }_{1}\mathbf{X}\right)\left\|\mathbf{X}\right\|^{\nu_{1}}\*A_{\nu_{2}}\left(% \mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)\left\|\mathbf{T}-\mathbf{X}\right\|% ^{\nu_{2}}\,\mathrm{d}{\mathbf{X}}=\left\|\mathbf{T}\right\|^{\nu_{1}+\nu_{2}+% \frac{1}{2}(m+1)}A_{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}\left((\mathbf{S}_{1}+% \mathbf{S}_{2})\mathbf{T}\right),$
$\nu_{j}\in\mathbb{C}$ , $\Re\left(\nu_{j}\right)>-1$ , $j=1,2$ ; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$ ; $\mathbf{T}\in{\boldsymbol{\Omega}}$ .
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\mathbb{C}$ : complex plane, $\in$ : element of, $\int$ : integral, $\Re$ : real part, $\boldsymbol{\mathcal{S}}$ : 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, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $<$ : less-than for matrices, $j$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ 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		$B_{\nu}\left(\mathbf{T}\right)=\left\|\mathbf{T}\right\|^{-\nu}B_{-\nu}\left(% \mathbf{T}\right),$
$\nu\in\mathbb{C}$ , $\mathbf{T}\in{\boldsymbol{\Omega}}$ .
ⓘ Symbols: $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (second kind), $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ and ${\boldsymbol{\Omega}}$ : 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		$\int_{\boldsymbol{\Omega}}A_{\nu_{1}}\left(\mathbf{T}\mathbf{X}\right)B_{-\nu_% {2}}\left(\mathbf{S}\mathbf{X}\right)\left\|\mathbf{X}\right\|^{\nu_{1}}\,% \mathrm{d}{\mathbf{X}}=\frac{1}{A_{\nu_{1}+\nu_{2}}\left(\boldsymbol{{0}}% \right)}\left\|\mathbf{S}\right\|^{\nu_{2}}\left\|\mathbf{T}+\mathbf{S}\right\|^{-% (\nu_{1}+\nu_{2}+\frac{1}{2}(m+1))},$
$\Re\left(\nu_{1}+\nu_{2}\right)>-1$ ; $\mathbf{S},\mathbf{T}\in{\boldsymbol{\Omega}}$ .
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (second kind), $\in$ : element of, $\int$ : integral, $\Re$ : 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, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ 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		$\int_{\mathbf{O}(m)}\operatorname{etr}\left(\mathbf{S}\mathbf{H}\right)\mathrm% {d}{\mathbf{H}}=\frac{A_{-1/2}\left(-\frac{1}{4}\mathbf{S}\mathbf{S}^{\mathrm{% T}}\right)}{A_{-1/2}\left(\boldsymbol{{0}}\right)},$
$\mathbf{S}$ arbitrary.
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : 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 $\boldsymbol{{0}}$ : $m\times m$ 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

§35.5(iii) Asymptotic Approximations

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Keywords:: Bessel functions, asymptotic approximations, of matrix argument
Permalink:: http://dlmf.nist.gov/35.5.iii
See also:: Annotations for §35.5 and Ch.35

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

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

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