Mehler–Dirichlet formula

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1: 14.31 Other Applications

… ►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). ►

§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: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

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§25.15(i) Definitions and Basic Properties

►The notation $L(s,\chi )$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … … ►

§25.15(ii) Zeros

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3: 14.12 Integral Representations

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§14.12(i)

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Mehler–Dirichlet Formula

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4: 27.5 Inversion Formulas

§27.5 Inversion Formulas

►If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates …called the Dirichlet product (or convolution) of $f$ and $g$. …which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … ►Other types of Möbius inversion formulas include: …

5: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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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).

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How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.

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External Links:

ISSN 0006-3835, Document, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§3.5(v), §9.17(iii).

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

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6: 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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7: 18.10 Integral Representations

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

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8: 25.19 Tables

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

9: 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).

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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.

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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:

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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).

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General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.

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External Links:

ISSN 0080-4614, FullText, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v), (9.13.18), §9.13(i), §9.13(i).

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10: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►In other words, Dirichlet characters (mod $k$) satisfy the four conditions: … ►If $\chi$ is a character (mod $k$), so is its complex conjugate $\overline{\chi }$. … ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if … ►Every Dirichlet character $\chi$ (mod $k$) is a product …