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functions of matrix argument

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11: 35.12 Software
  • Koev and Edelman (2006). Computation of hypergeometric functions of matrix argument in MATLAB.

  • 12: Errata
  • Other Changes


    • In Subsection 1.9(i), just below (1.9.1), a phrase was added which elaborates that i 2 = - 1 .

    • Poor spacing in math was corrected in several chapters.

    • In Section 1.13, there were several modifications. In Equation (1.13.4), the determinant form of the two-argument Wronskian

      1.13.4
      𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) - w 2 ( z ) w 1 ( z )

      was added as an equality. In Paragraph Wronskian in §1.13(i), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j - 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.

    • In Section 3.1, there were several modifications. In Paragraph IEEE Standard in §3.1(i), the description was modified to reflect the most recent IEEE 754-2019 Floating-Point Arithmetic Standard IEEE (2019). In the new standard, single, double and quad floating-point precisions are replaced with new standard names of binary32, binary64 and binary128. Figure 3.1.1 has been expanded to include the binary128 floating-point memory positions and the caption has been updated using the terminology of the 2019 standard. A sentence at the end of Subsection 3.1(ii) has been added referring readers to the IEEE Standards for Interval Arithmetic IEEE (2015, 2018). This was suggested by Nicola Torracca.

    • In Equation (35.7.3), originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument F 1 2 was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

  • Other Changes


  • 13: Bibliography R
  • D. St. P. Richards (2004) Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.
  • 14: Bibliography H
  • C. S. Herz (1955) Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.
  • 15: Bibliography W
  • G. Wei and B. E. Eichinger (1993) Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.
  • 16: Software Index
    17: Bibliography B
  • R. W. Butler and A. T. A. Wood (2002) Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.
  • R. W. Butler and A. T. A. Wood (2003) Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.
  • 18: Bibliography G
  • K. I. Gross and D. St. P. Richards (1987) Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.
  • 19: Bibliography K
  • P. Koev and A. Edelman (2006) The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75 (254), pp. 833–846.
  • T. H. Koornwinder and I. Sprinkhuizen-Kuyper (1978) Hypergeometric functions of 2 × 2 matrix argument are expressible in terms of Appel’s functions F 4 . Proc. Amer. Math. Soc. 70 (1), pp. 39–42.
  • 20: Bibliography O
  • I. Olkin (1959) A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.
  • F. W. J. Olver (1951) A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.
  • F. W. J. Olver (1980b) Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.
  • K. Ono (2000) Distribution of the partition function modulo m . Ann. of Math. (2) 151 (1), pp. 293–307.
  • C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.