generalized Mehler–Fock transformation

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41: 14.29 Generalizations

§14.29 Generalizations

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14.29.1

\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+{\left(\nu(\nu+1)-\frac{\mu_{1}^{2}}{2(1-z)}-\frac{% \mu_{2}^{2}}{2(1+z)}\right)w}=0

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.29
Permalink:: http://dlmf.nist.gov/14.29.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.29 and Ch.14

►are called Generalized Associated Legendre Functions. … …

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute

\pi

to high precision (Borwein and Borwein (1987, p. 26)). …

43: 18.2 General Orthogonal Polynomials

§18.2 General Orthogonal Polynomials

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Orthogonality on General Sets

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§18.2(vii) Quadratic Transformations

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Generalizations of the Szegő Class

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44: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

… ►can be viewed as a generalization of (12.2.4). …

45: 14.17 Integrals

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14.17.1

{\int\left(1-x^{2}\right)^{-\mu/2}\mathsf{P}^{\mu}_{\nu}\left(x\right)\,% \mathrm{d}x}={-\left(1-x^{2}\right)^{-(\mu-1)/2}\mathsf{P}^{\mu-1}_{\nu}\left(% x\right)}.

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.17(i)
Permalink:: http://dlmf.nist.gov/14.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.17(i), §14.17 and Ch.14

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§14.17(v) Laplace Transforms

►For Laplace transforms and inverse Laplace transforms involving associated Legendre functions, see Erdélyi et al. (1954a, pp. 179–181, 270–272), Oberhettinger and Badii (1973, pp. 113–118, 317–324), Prudnikov et al. (1992a, §§3.22, 3.32, and 3.33), and Prudnikov et al. (1992b, §§3.20, 3.30, and 3.31). ►

§14.17(vi) Mellin Transforms

►For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).

46: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

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§9.13(i) Generalizations from the Differential Equation

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§9.13(ii) Generalizations from Integral Representations

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47: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing

\operatorname{Ai}\left(z\right)

in the complex plane, once values of this function can be generated on the positive real axis. For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …

48: 10.74 Methods of Computation

… ►For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions

K_{\nu}\left(z\right)

for

0<\nu<1

and

0<x<\infty

see Gautschi (2002a). … ►Then

J_{n}\left(x\right)

and

Y_{n}\left(x\right)

can be generated by either forward or backward recurrence on

n

when

n<x

, but if

n>x

then to maintain stability

J_{n}\left(x\right)

has to be generated by backward recurrence on

n

, and

Y_{n}\left(x\right)

has to be generated by forward recurrence on

n

. … ►

Hankel Transform

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Spherical Bessel Transform

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Kontorovich–Lebedev Transform

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49: 16.23 Mathematical Applications

§16.23 Mathematical Applications

… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … ►

§16.23(ii) Random Graphs

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§16.23(iv) Combinatorics and Number Theory

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50: 18.18 Sums

… ►See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of

P^{(\gamma,\delta)}_{n}\left(x\right)

in terms of

P^{(\alpha,\beta)}_{n}\left(x\right)

. … ►See (18.2.41) for the Poisson kernel in case of general OP’s. … ►

Hermite

… ►Formula (18.18.28) is known as the Mehler formula. See Ismail (2009, Theorem 4.7.2) for a formula called Kibble–Slepian formula, which generalizes (18.18.28). …