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In Gauss quadrature (also known as GaussChristoffel quadrature) we use (3.5.15) with nodes $x_{k}$ the zeros of $p_{n}$, and weights $w_{k}$ given by …The remainder is given by …
##### 2: 35.10 Methods of Computation
Other methods include numerical quadrature applied to double and multiple integral representations. 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). …
##### 3: 15.19 Methods of Computation
The Gauss series (15.2.1) converges for $|z|<1$. … Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. For fast computation of $F\left(a,b;c;z\right)$ with $a,b$ and $c$ complex, and with application to Pöschl–Teller-Ginocchio potential wave functions, see Michel and Stoitsov (2008). … Gauss quadrature approximations are discussed in Gautschi (2002b). … For example, in the half-plane $\Re z\leq\frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F\left(a,b;c+N+1;z\right)$ and $F\left(a,b;c+N;z\right)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …
##### 4: 9.17 Methods of Computation
For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … The second method is to apply generalized Gauss–Laguerre quadrature3.5(v)) to the integral (9.5.8). … For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983). …
##### 5: 15.5 Derivatives and Contiguous Functions
The six functions $F\left(a\pm 1,b;c;z\right)$, $F\left(a,b\pm 1;c;z\right)$, $F\left(a,b;c\pm 1;z\right)$ are said to be contiguous to $F\left(a,b;c;z\right)$.
15.5.11 $(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c;z\right)+a(z% -1)F\left(a+1,b;c;z\right)=0,$
15.5.12 $(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1;c;z\right)=0,$
15.5.13 $(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;z\right)-(c-b)F\left(a,b-1;c% ;z\right)=0,$
By repeated applications of (15.5.11)–(15.5.18) any function $F\left(a+k,b+\ell;c+m;z\right)$, in which $k,\ell,m$ are integers, can be expressed as a linear combination of $F\left(a,b;c;z\right)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. …
##### 6: 6.18 Methods of Computation
Quadrature of the integral representations is another effective method. For example, the Gauss-Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss-Legendre formula (§3.5(v)) see Tooper and Mark (1968). … Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. …
##### 7: Bibliography G
• C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).
• W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
• W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
• W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
• G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
• ##### 8: 15.3 Graphics Figure 15.3.1: F ⁡ ( 4 3 , 9 16 ; 14 5 ; x ) , - 100 ≤ x ≤ 1 . Magnify Figure 15.3.2: F ⁡ ( 5 , - 10 ; 1 ; x ) , - 0.023 ≤ x ≤ 1 . Magnify Figure 15.3.3: F ⁡ ( 1 , - 10 ; 10 ; x ) , - 3 ≤ x ≤ 1 . Magnify Figure 15.3.4: F ⁡ ( 5 , 10 ; 1 ; x ) , - 1 ≤ x ≤ 0.022 . Magnify Figure 15.3.5: F ⁡ ( 4 3 , 9 16 ; 14 5 ; x + i ⁢ y ) , 0 ≤ x ≤ 2 , - 0.5 ≤ y ≤ 0.5 . … Magnify 3D Help
##### 9: 16.12 Products
16.12.1 ${{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\right).$
16.12.2 $\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}};z\right)\right)^{2}={{}_{3% }F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b};z\right).$
16.12.3 $\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},$ $|z|<1$.
##### 10: Bibliography T
• N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.
• N. M. Temme (2003) Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
• L. N. Trefethen (2008) Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.
• L. N. Trefethen (2011) Six myths of polynomial interpolation and quadrature. Math. Today (Southend-on-Sea) 47 (4), pp. 184–188.