quadratic transformations

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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

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§15.8(iii) Quadratic Transformations

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Table 15.8.1: Quadratic transformations of the hypergeometric function.

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Group 1	Group 2	Group 3	Group 4
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§15.8(iv) Quadratic Transformations (Continued)

… ►This is a quadratic transformation between two cases in Group 1. … ►which is a quadratic transformation between two cases in Group 3. …

3: 18.7 Interrelations and Limit Relations

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§18.7(ii) Quadratic Transformations

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4: 19.8 Quadratic Transformations

§19.8 Quadratic Transformations

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5: 19.36 Methods of Computation

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§19.36(ii) Quadratic Transformations

►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 $R_F(x,y,z)$ and $R_G(x,y,z)$ 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)). …

6: 32.7 Bäcklund Transformations

… ► ${\text{P}}_{\text{VI}}$ also has quadratic and quartic transformations. …The quadratic transformation …

7: 16.6 Transformations of Variable

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Quadratic

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8: 18.2 General Orthogonal Polynomials

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§18.2(vii) Quadratic Transformations

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9: 16.16 Transformations of Variables

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16.16.10 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\beta -\alpha \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{(-y)}^{-\alpha }{F}_4(\alpha ,\alpha -{\gamma }^{\prime }+1;\gamma ,\alpha -\beta +1;\frac{x}{y},\frac{1}{y})+\frac{\mathrm{\Gamma }\left({\gamma }^{\prime }\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}{\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)\mathrm{\Gamma }\left(\alpha \right)}{(-y)}^{-\beta }{F}_4(\beta ,\beta -{\gamma }^{\prime }+1;\gamma ,\beta -\alpha +1;\frac{x}{y},\frac{1}{y}).$$

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Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Permalink:

http://dlmf.nist.gov/16.16.E10

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See also:

Annotations for §16.16(ii), §16.16 and Ch.16

►For quadratic transformations of Appell functions see Carlson (1976). …

10: 19.22 Quadratic Transformations

§19.22 Quadratic Transformations

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