(For other notation see Notation for the Special Functions.)
All matrices are of order , unless specified otherwise. All fractional or complex powers are principal values.
complex variables. |
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nonnegative integers. |
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positive integer. |
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partitional shifted factorial (§35.4(i)). |
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zero matrix. |
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identity matrix. |
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space of all real symmetric matrices. |
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real symmetric matrices. |
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trace of . |
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. |
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determinant of (except when where it means either determinant or absolute value, depending on the context). |
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th principal minor of . |
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th element of . |
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. |
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space of positive-definite real symmetric matrices. |
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eigenvalues of . |
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spectral norm of . |
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is positive definite. Similarly, is equivalent. |
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complex symmetric matrix. |
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complex-valued function with . |
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space of orthogonal matrices. |
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orthogonal matrix. |
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normalized Haar measure on . |
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zonal polynomials. |
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric or ; generalized hypergeometric or .