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##### 1: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …
##### 2: 15.8 Transformations of Variable
This is a quadratic transformation between two cases in Group 1. … which is a quadratic transformation between two cases in Group 3. …
##### 4: 19.36 Methods of Computation
Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. The incomplete integrals $R_{F}\left(x,y,z\right)$ and $R_{G}\left(x,y,z\right)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_{C}$, accompanied by two quadratically convergent series in the case of $R_{G}$; compare Carlson (1965, §§5,6). … Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …
##### 5: 32.7 Bäcklund Transformations
$\mbox{P}_{\mbox{\scriptsize VI}}$ also has quadratic and quartic transformations. …The quadratic transformation
##### 6: 16.6 Transformations of Variable
16.16.10 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\frac{\Gamma\left(% \gamma^{\prime}\right)\Gamma\left(\beta-\alpha\right)}{\Gamma\left(\gamma^{% \prime}-\alpha\right)\Gamma\left(\beta\right)}(-y)^{-\alpha}{F_{4}}\left(% \alpha,\alpha-\gamma^{\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}% \right)+\frac{\Gamma\left(\gamma^{\prime}\right)\Gamma\left(\alpha-\beta\right% )}{\Gamma\left(\gamma^{\prime}-\beta\right)\Gamma\left(\alpha\right)}(-y)^{-% \beta}{F_{4}}\left(\beta,\beta-\gamma^{\prime}+1;\gamma,\beta-\alpha+1;\frac{x% }{y},\frac{1}{y}\right).$