# Gaussian hypergeometric function

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##### 2: 35.10 Methods of Computation
See Yan (1992) for the ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $\mathbf{O}(m)$ applied to a generalization of the integral (35.5.8). …
##### 4: 35.1 Special Notation
 $a,b$ complex variables. …
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively $\Gamma_{m}\left(a\right)$ and $\mathrm{B}_{m}\left(a,b\right)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu}\left(\mathbf{T}\right)$ and (of the second kind) $B_{\nu}\left(\mathbf{T}\right)$; confluent hypergeometric (of the first kind) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ or $\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\Psi\left(a;b;\mathbf{T}\right)$; Gaussian hypergeometric ${{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$; generalized hypergeometric ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. … Related notations for the Bessel functions are $\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)$ (Terras (1988, pp. 49–64)), and $\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)$ (Faraut and Korányi (1994, pp. 357–358)).
##### 5: Bibliography H
• Y. P. Hsu (1993) Development of a Gaussian hypergeometric function code in complex domains. Internat. J. Modern Phys. C 4 (4), pp. 805–840.
• ##### 6: 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.

• ##### 7: Bibliography G
• L. Gatteschi (1990) New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.
• W. Gautschi (1983) How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.
• W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.
• W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
• K. I. Gross and D. St. P. Richards (1991) Hypergeometric functions on complex matrix space. Bull. Amer. Math. Soc. (N.S.) 24 (2), pp. 349–355.
• ##### 8: Bibliography S
• J. B. Seaborn (1991) Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.
• I. Shavitt (1963) The Gaussian Function in Calculations of Statistical Mechanics and Quantum Mechanics. In Methods in Computational Physics: Advances in Research and Applications, B. Alder, S. Fernbach, and M. Rotenberg (Eds.), Vol. 2, pp. 1–45.
• G. Shimura (1982) Confluent hypergeometric functions on tube domains. Math. Ann. 260 (3), pp. 269–302.
• L. J. Slater (1966) Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.
• A. H. Stroud and D. Secrest (1966) Gaussian Quadrature Formulas. Prentice-Hall Inc., Englewood Cliffs, N.J..
• ##### 9: Bibliography F
• S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.
• J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.
• R. C. Forrey (1997) Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.
• B. D. Fried and S. D. Conte (1961) The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. Academic Press, London-New York.
• Y. V. Fyodorov (2005) Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.
• ##### 10: Bibliography C
• L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.
• B. C. Carlson (1985) The hypergeometric function and the $R$-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.
• B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.
• F. Chapeau-Blondeau and A. Monir (2002) Numerical evaluation of the Lambert $W$ function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.
• S. Conde and S. L. Kalla (1981) On zeros of the hypergeometric function. Serdica 7 (3), pp. 243–249.