# functions of matrix argument

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## 11—20 of 25 matching pages

##### 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 ${\mathrm{i}^{2}}=-1$.

• 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
$\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)$

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 $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, where $1\leq j,k\leq n$. 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 ${{}_{2}F_{1}}$ 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.
##### 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 $2\times 2$ 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.