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DLMF: §35.6 Confluent Hypergeometric Functions of Matrix Argument ‣ Properties ‣ Chapter 35 Functions of Matrix Argument

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35 Functions of Matrix Argument Properties 35.5 Bessel Functions of Matrix Argument 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

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Keywords:: confluent hypergeometric functions, of matrix argument
Permalink:: http://dlmf.nist.gov/35.6
See also:: Annotations for Ch.35

Contents

§35.6(i) Definitions

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Keywords:: confluent hypergeometric functions, definition, of matrix argument
Notes:: See Koecher (1954), Muirhead (1982, pp. 472–473). See also Herz (1955), Muirhead (1978).
Permalink:: http://dlmf.nist.gov/35.6.i
See also:: Annotations for §35.6 and Ch.35

35.6.1		${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!% }\sum_{\|\kappa\|=k}\frac{{\left[a\right]_{\kappa}}}{{\left[b\right]_{\kappa}}}Z% _{\kappa}\left(\mathbf{T}\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : 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		$\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)\left\|% \mathbf{X}\right\|^{a-\frac{1}{2}(m+1)}\*{\left\|\mathbf{I}+\mathbf{X}\right\|}^{% b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$
$\Re\left(a\right)>\frac{1}{2}(m-1)$ , $\mathbf{T}\in{\boldsymbol{\Omega}}$ .
ⓘ Defines: $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind) Symbols: $\in$ : element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : 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, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : 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

Laguerre Form

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Keywords:: Laguerre form, confluent hypergeometric functions, of matrix argument
See also:: Annotations for §35.6(i), §35.6 and Ch.35

35.6.3		$L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+\nu+% \frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*{{}_% {1}F_{1}}\left({-\nu\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),$
$\Re\left(\gamma\right),\Re\left(\gamma+\nu\right)>-1$ .
ⓘ Defines: $L^{(\NVar{\gamma})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Laguerre function of matrix argument Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.6.E3 Encodings: TeX, pMML, png See also: Annotations for §35.6(i), §35.6(i), §35.6 and Ch.35

§35.6(ii) Properties

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Keywords:: confluent hypergeometric functions, of matrix argument, properties
Notes:: See Herz (1955), Muirhead (1982, pp. 264–266, 472–473). For (35.6.6) apply (35.2.3) and (35.6.5). See also Shimura (1982).
Permalink:: http://dlmf.nist.gov/35.6.ii
See also:: Annotations for §35.6 and Ch.35

35.6.4		${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\operatorname{% etr}\left(\mathbf{T}\mathbf{X}\right)\left\|\mathbf{X}\right\|^{a-\frac{1}{2}(m+% 1)}\left\|\mathbf{I}-\mathbf{X}\right\|^{b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$
$\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : 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, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ 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		$\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left\|\mathbf{X}\right\|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};% \mathbf{S}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(b\right)% \left\|\mathbf{I}-\mathbf{S}\mathbf{T}^{-1}\right\|^{-a}\left\|\mathbf{T}\right\|^% {-b},$
$\mathbf{T}>\mathbf{S}$ , $\Re\left(b\right)>\frac{1}{2}(m-1)$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : 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, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable, $>$ : 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		$\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left\|\mathbf{T}\right\|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left\|\mathbf{X}\right\|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left\|\mathbf{T}-\mathbf{X}\right\|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}},$
$\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{T}$ : 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, $b$ : complex variable, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ 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		${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\operatorname{etr}\left(% \mathbf{T}\right){{}_{1}F_{1}}\left({b-a\atop b};-\mathbf{T}\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : 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		$\int_{\boldsymbol{\Omega}}\left\|\mathbf{T}\right\|^{c-\frac{1}{2}(m+1)}\Psi% \left(a;b;\mathbf{T}\right)\,\mathrm{d}{\mathbf{T}}=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{% \Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},$
$\Re\left(a\right)>\Re\left(c\right)+\frac{1}{2}(m-1)>m-1$ , $\Re\left(c-b\right)>-1$ .
ⓘ Symbols: $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{T}$ : 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, $b$ : 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(iii) Relations to Bessel Functions of Matrix Argument

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Keywords:: Bessel functions, confluent hypergeometric functions, of matrix argument, relations to Bessel functions of matrix argument, relations to confluent hypergeometric functions of matrix argument
Notes:: See Herz (1955), Muirhead (1978).
Permalink:: http://dlmf.nist.gov/35.6.iii
See also:: Annotations for §35.6 and Ch.35

35.6.9		$\lim_{a\to\infty}{{}_{1}F_{1}}\left({a\atop\nu+\frac{1}{2}(m+1)};-a^{-1}% \mathbf{T}\right)=\frac{A_{\nu}\left(\mathbf{T}\right)}{A_{\nu}\left(% \boldsymbol{{0}}\right)}.$
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ 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		$\lim_{a\to\infty}\Gamma_{m}\left(a\right)\Psi\left(a+\nu;\nu+\tfrac{1}{2}(m+1)% ;a^{-1}\mathbf{T}\right)=B_{\nu}\left(\mathbf{T}\right).$
ⓘ Symbols: $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (second kind), $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : 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

§35.6(iv) Asymptotic Approximations

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Keywords:: asymptotic approximations, confluent hypergeometric functions, of matrix argument
Permalink:: http://dlmf.nist.gov/35.6.iv
See also:: Annotations for §35.6 and Ch.35

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

35.5 Bessel Functions of Matrix Argument 35.7 Gaussian Hypergeometric Function of Matrix Argument

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