Mehler functions

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1: 14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

2: 14.31 Other Applications

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§14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …

3: 14.12 Integral Representations

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§14.12(i)

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4: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

… ►Solutions are known as conical or Mehler functions. … ►

§14.20(vi) Generalized Mehler–Fock Transformation

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5: 14.34 Software

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§14.34(iv) Conical (Mehler) and/or Toroidal Functions

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6: 10.9 Integral Representations

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Mehler–Sonine and Related Integrals

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7: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

… ►See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions. ►

Ultraspherical

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Hermite

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§18.11(ii) Formulas of Mehler–Heine Type

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8: 18.18 Sums

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§18.18(i) Series Expansions of Arbitrary Functions

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Expansion of $L^2$ functions

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Laguerre

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Hermite

… ►Formula (18.18.28) is known as the Mehler formula. …

9: 18.10 Integral Representations

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§18.10(i) Dirichlet–Mehler-Type Integral Representations

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18.10.5 $$P_n\left(\cos \theta \right)=\frac{1}{\pi }{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^{n}d\varphi .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

A&S Ref:

22.10.12

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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18.10.6 $$L_n^{(\alpha )}\left(x^2\right)=\frac{2{(-1)}^n}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)n!}{\int }_0^{\mathrm{\infty }}{\int }_0^{\pi }{(x^2-r^2+2ixr\cos \varphi )}^n{\mathrm{e}}^{-r^2}{r}^{2\alpha +1}{(\sin \varphi )}^{2\alpha }d\varphi dr,$$ $\alpha >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

… ►

Laguerre

… ►For the Bessel function $J_{\nu }\left(z\right)$ see §10.2(ii). …

10: Bibliography G

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

… ►

Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.

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External Links:

ZentralBlatt

Referenced by:

§28.26(i), §28.26(i), §28.8(iii).

Encodings:

BibTeX

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Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

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External Links:

Document

Referenced by:

§19.35(ii).

Encodings:

BibTeX