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: 37.7 Parabolic Biangular Region with Weight Function $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

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§37.7(ii) Quadratic Transformations

►The Jacobi polynomials (37.3.3) on

\triangle

and the Jacobi polynomials (37.7.3) on

\mathbb{A}

are related by the quadratic transformations … ►The Jacobi polynomials (37.7.3) on

\mathbb{A}

and the ultraspherical polynomials (37.4.4) on

\mathbb{D}

are related by the quadratic transformations … ►The Jacobi polynomials (37.7.3) on

\mathbb{A}

are related to the real disk polynomials (37.4.15) by the quadratic transformations … ►The polynomials (37.3.9) and (37.7.16) are related by the quadratic transformations …

3: 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.

► ►►

Group 1	Group 2	Group 3	Group 4
…

► … ►

§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. …

4: 18.7 Interrelations and Limit Relations

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§18.7(ii) Quadratic Transformations

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5: 19.8 Quadratic Transformations

§19.8 Quadratic Transformations

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6: 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}\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)). …

7: 32.7 Bäcklund Transformations

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\mbox{P}_{\mbox{\scriptsize VI}}

also has quadratic and quartic transformations. …The quadratic transformation …

8: 16.6 Transformations of Variable

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Quadratic

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9: 37.12 Orthogonal Polynomials on Quadratic Surfaces

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§37.12(iv) Quadratic Transformations

… ►Then there are quadratic transformations for the polynomials (37.12.9) and (37.12.14) in terms of complex disk polynomials (37.4.11) and complex circular Hermite polynomials (37.6.3), respectively: … ►

S^{n}_{\ell,m}\left(z^{2},|{z}|^{2};0\right)=\begin{cases}z^{-1}S_{n+m+1,n-m}% \left(z,\overline{z}\right),&Y_{\ell}^{m}({\mathrm{e}}^{\mathrm{i}\theta})={% \mathrm{e}}^{\mathrm{i}m\theta},\\ {\overline{z}}^{-1}S_{n-m,n+m+1}\left(z,\overline{z}\right),&Y_{\ell}^{m}({% \mathrm{e}}^{\mathrm{i}\theta})={\mathrm{e}}^{-\mathrm{i}m\theta},\end{cases}

0\leq m\leq n

z\in\mathbb{C}

10: 18.2 General Orthogonal Polynomials

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§18.2(vii) Quadratic Transformations

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