quadratic transformations

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11: Bibliography W

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Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.

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ISSN 1687-1847, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§16.6, §16.6.

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12: Bibliography C

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Quadratic transformations of Appell functions. SIAM J. Math. Anal. 7 (2), pp. 291–304.

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Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.16(ii), §19.22(i).

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An extension of a Kummer’s quadratic transformation formula with an application. Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.

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ISSN 1598-7264, MathReview (Petr Blaschke), ZentralBlatt

Referenced by:

§16.6, §16.6.

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13: 15.9 Relations to Other Functions

… ►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. …

14: 18.33 Polynomials Orthogonal on the Unit Circle

… ►After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi }_n$. …

15: 31.7 Relations to Other Functions

… ►They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). …

16: Errata

… ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. …

17: 10.74 Methods of Computation

… ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. … ►

Hankel Transform

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Spherical Bessel Transform

►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when $\nu$ is half an odd positive integer. … ►

Kontorovich–Lebedev Transform

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18: Bibliography M

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§13.12.

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Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

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ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt

Referenced by:

§2.6(ii).

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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Bäcklund transformations and solution hierarchies for the third Painlevé equation. Stud. Appl. Math. 98 (2), pp. 139–194.

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ISSN 0022-2526, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.10(iii), §32.7(iii), §32.8(iii), §32.9(i).

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19: 2.3 Integrals of a Real Variable

… ►Assume that the Laplace transform …Then … ►A uniform approximation can be constructed by quadratic change of integration variable: … ►The integral (2.3.24) transforms into … ►

§2.3(vi) Asymptotics of Mellin Transforms

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20: 19.14 Reduction of General Elliptic Integrals

… ►In (19.14.4) , each quadratic polynomial is positive on the interval $(y,x)$, and $\alpha ,\beta ,\gamma$ is a permutation of $0,a_1{b}_2,a_2{b}_1$ (not all 0 by assumption) such that $\alpha \le \beta \le \gamma$. … ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. …The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …