# Gauss transformation

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##### 1: 19.15 Advantages of Symmetry
Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … …
If $x,y,z$ are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when $x,y (implying $a), and descending Gauss transformations when $z (implying $z_{+}). …Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. …
###### §19.8(iii) GaussTransformation
We consider only the descending Gauss transformation because its (ascending) inverse moves $F\left(\phi,k\right)$ closer to the singularity at $k=\sin\phi=1$. …
##### 5: 19.36 Methods of Computation
The step from $n$ to $n+1$ is an ascending Landen transformation if $\theta=1$ (leading ultimately to a hyperbolic case of $R_{C}$) or a descending Gauss transformation if $\theta=-1$ (leading to a circular case of $R_{C}$). … Descending Gauss transformations of $\Pi\left(\phi,\alpha^{2},k\right)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … The function $\mathrm{el2}\left(x,k_{c},a,b\right)$ is computed by descending Landen transformations if $x$ is real, or by descending Gauss transformations if $x$ is complex (Bulirsch (1965b)). …
##### 6: 16.6 Transformations of Variable
16.6.2 ${{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).$
For Kummer-type transformations of ${{}_{2}F_{2}}$ functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
##### 7: Bibliography F
• H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gausstransformation. Math. Comp. 19 (89), pp. 97–104.
• ##### 8: Bibliography V
• R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
• ##### 9: 16.4 Argument Unity
Balanced ${{}_{4}F_{3}}\left(1\right)$ series have transformation formulas and three-term relations. … Transformations for both balanced ${{}_{4}F_{3}}\left(1\right)$ and very well-poised ${{}_{7}F_{6}}\left(1\right)$ are included in Bailey (1964, pp. 56–63). …
##### 10: 16.5 Integral Representations and Integrals
16.5.2 ${{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=% \frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}\right)\Gamma\left(b_{0}-a_{0% }\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0}-1}{{}_{p}F_{q}}\left({a_{1}% ,\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)\mathrm{d}t,$ $\Re b_{0}>\Re a_{0}>0$,
16.5.3 ${{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}^{-t}}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\mathrm{d}t,$ $\Re z<1$, $\Re a_{0}>0$,
16.5.4 ${{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=\frac% {\Gamma\left(b_{0}\right)}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm% {i}\infty}{\mathrm{e}^{t}}t^{-b_{0}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};\frac{z}{t}\right)\mathrm{d}t,$ $c>0$, $\Re b_{0}>0$.