Mehler–Heine type formulas

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1: 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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2: 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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3: 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. …

4: 14.28 Sums

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§14.28(ii) Heine’s Formula

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5: 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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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

Encodings:

BibTeX

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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

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External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

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6: 18.7 Interrelations and Limit Relations

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

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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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

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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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Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. J. Comput. Appl. Math. 153 (1-2), pp. 225–234.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.45, §10.74(viii).

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: 14.20 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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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. …