Mehler%E2%80%93Heine%20type%20formulas

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1: 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. …

2: 18.11 Relations to Other Functions

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§18.11(i) Explicit Formulas

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§18.11(ii) Formulas of Mehler–Heine Type

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3: Bibliography O

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Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:

§1.14(viii).

Encodings:

BibTeX

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Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

Encodings:

BibTeX

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Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

BibTeX

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.

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External Links:

ISSN 0923-2958, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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4: 14.12 Integral Representations

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§14.12(i)

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Mehler–Dirichlet Formula

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Heine’s Integral

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5: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

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

14.20.2 $${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\mathrm{\Re }\left({\mathrm{e}}^{\mu \pi \mathrm{i}}{𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)\right)-\frac{1}{2}\pi \sin \left(\mu \pi \right){𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right).$$

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Defines:

${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function

Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i), §14.23

Permalink:

http://dlmf.nist.gov/14.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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14.20.6 $$P_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\mathrm{i}{\mathrm{e}}^{-\mu \pi \mathrm{i}}}{\sinh \left(\tau \pi \right){\left|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)\right|}^2}\left(Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)-Q_{-\frac{1}{2}-\mathrm{i}\tau }^{\mu }\left(x\right)\right),$$ $\tau \ne 0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i)

Permalink:

http://dlmf.nist.gov/14.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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§14.20(vi) Generalized Mehler–Fock Transformation

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6: 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). …

7: 18.10 Integral Representations

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

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§18.10(ii) Laplace-Type Integral Representations

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8: 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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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

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9: 14.34 Software

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

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10: 18.18 Sums

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§18.18(v) Linearization Formulas

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§18.18(vi) Bateman-Type Sums

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Jacobi

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Hermite

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