Mehler%E2%80%93Dirichlet%20formula

SearchAdvancedHelp

(0.001 seconds)

21—30 of 330 matching pages

21: Bibliography M

… ►

Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

ⓘ

External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

Encodings:

BibTeX

… ►

On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§14.17(i).

Encodings:

BibTeX

… ►

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

ⓘ

External Links:

ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

►

Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

ⓘ

External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

… ►

On the representation of numbers as a sum of $2r$ squares. Quarterly Journal of Math. 48, pp. 93–104.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

…

22: 36.5 Stokes Sets

… ►

§36.5(ii) Cuspoids

… ►

36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

ⓘ

Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►

36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

ⓘ

Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►

§36.5(iii) Umbilics

…

23: 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 $g$. The set of all number-theoretic functions $f$ with $f(1)\ne 0$ forms an abelian group under Dirichlet multiplication, with the function $\lfloor 1/n\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). … ►Other types of Möbius inversion formulas include: …

24: 32.8 Rational Solutions

… ►

32.8.3 $$w(z;3)=\frac{3z^2}{z^3+4}-\frac{6z^2(z^3+10)}{z^6+20z^3-80},$$

ⓘ

Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.8(ii), §32.8 and Ch.32

►

32.8.4 $$w(z;4)=-\frac{1}{z}+\frac{6z^2(z^3+10)}{z^6+20z^3-80}-\frac{9z^5(z^3+40)}{z^9+60z^6+11200}.$$

ⓘ

Symbols:

$z$: real

Permalink:

http://dlmf.nist.gov/32.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §32.8(ii), §32.8 and Ch.32

… ►

$$Q_3(z)=z^6+20z^3-80,$$

…

25: 26.13 Permutations: Cycle Notation

…

26: 26.6 Other Lattice Path Numbers

… ►

Table 26.6.1: Delannoy numbers $D(m,n)$.

► ►►►

$m$	$n$
	…
10	1	21	221	1561	8361	36365	1 34245	4 33905	12 56465	33 17445	80 97453

►

… ►

Table 26.6.3: Narayana numbers $N(n,k)$.

► ►►►

$n$	$k$
	…
5	0	1	10	20	10	1
…

►

…

27: 3.4 Differentiation

… ►

Two-Point Formula

… ►

Three-Point Formula

… ►

Four-Point Formula

… ►

Five-Point Formula

… ►

Six-Point Formula

…

28: 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 …

29: 25.19 Tables

… ►

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.

►

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.

30: Bibliography P

… ►

An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

ⓘ

Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

External Links:

Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

Encodings:

BibTeX

… ►

Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

… ►

Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.

ⓘ

External Links:

Document

Referenced by:

§10.76(ii).

Encodings:

BibTeX

… ►

Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.

ⓘ

External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.4.

Encodings:

BibTeX

… ►

Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§8.24(ii).

Encodings:

BibTeX

…