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1: 23.15 Definitions

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$k={\displaystyle \frac{{\theta }_2^2(0,q)}{{\theta }_3^2(0,q)}},$

… ►The set of all bilinear transformations of this form is denoted by SL $(2,\mathbb{Z})$ (Serre (1973, p. 77)). ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{<span class="ltx\_text hilite">SL</span>}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL $(2,\mathbb{Z})$, …(Some references refer to $2\mathrm{\ell }$ as the level). …

2: 15.17 Mathematical Applications

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL $(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. …

3: 23.18 Modular Transformations

… ►and $\lambda \left(\tau \right)$ is a cusp form of level zero for the corresponding subgroup of SL $(2,\mathbb{Z})$. … ► $J\left(\tau \right)$ is a modular form of level zero for SL $(2,\mathbb{Z})$. … ►

23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

Encodings:

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See also:

Annotations for §23.18, §23.18 and Ch.23

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23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$ $c>0$.

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , $\left(m,n\right)$: greatest common divisor (gcd), $c$: integer and $d$: integer

Referenced by:

§23.18, Erratum (V1.0.11) for Equation (23.18.7)

Permalink:

http://dlmf.nist.gov/23.18.E7

Encodings:

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Errata (effective with 1.0.11):

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare

See also:

Annotations for §23.18, §23.18 and Ch.23

… ►Note that $\eta \left(\tau \right)$ is of level $\frac{1}{2}$. …

4: 26.12 Plane Partitions

… ►The number of self-complementary plane partitions in $B(2r,2s,2t)$ is …in $B(2r+1,2s,2t)$ it is …in $B(2r+1,2s+1,2t)$ it is … ►The number of symmetric self-complementary plane partitions in $B(2r,2r,2t)$ is …in $B(2r+1,2r+1,2t)$ it is …

5: Bibliography F

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Tablicy značeniĭ funkcii $w(z)=e^{-z^2}(1+\frac{2i}{\sqrt{\pi }}{\int }_0^z{e}^{t^2}𝑑t)$ ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).

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Notes:

In Russian. English translation, see Faddeyeva and Terent’ev (1961)

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§7.21.

Encodings:

BibTeX

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

Encodings:

BibTeX

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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External Links:

ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:

ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

Encodings:

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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6: 8.6 Integral Representations

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8.6.2 $$\gamma (a,z)=z^{\frac{1}{2}a}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}a-1}{J}_a\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }a>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: parameter

Referenced by:

§8.6(i)

Permalink:

http://dlmf.nist.gov/8.6.E2

Encodings:

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See also:

Annotations for §8.6(i), §8.6 and Ch.8

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8.6.6 $$\mathrm{\Gamma }(a,z)=\frac{2z^{\frac{1}{2}a}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(1-a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\frac{1}{2}a}{K}_a\left(2\sqrt{zt}\right)dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: parameter

Referenced by:

§8.6(i)

Permalink:

http://dlmf.nist.gov/8.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(i), §8.6 and Ch.8

… ►In (8.6.10)–(8.6.12), $c$ is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at $s=a$, in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots }$. ►

8.6.10 $$\gamma (a,z)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s\right)}{a-s}{z}^{a-s}ds,$$ , $a\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter and $c$: real constant

Keywords:

Mellin transform

Referenced by:

§8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))

Permalink:

http://dlmf.nist.gov/8.6.E10

Encodings:

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See also:

Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

… ►For collections of integral representations of $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ see Erdélyi et al. (1953b, §9.3), Oberhettinger (1972, pp. 68–69), Oberhettinger and Badii (1973, pp. 309–312), Prudnikov et al. (1992b, §3.10), and Temme (1996b, pp. 282–283).

7: Bibliography L

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Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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External Links:

ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§30.12.

Encodings:

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Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:

ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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Orthogonal polynomials for exponential weights $x^{2\rho }{e}^{-2Q(x)}$ on $[0,d)$ . J. Approx. Theory 134 (2), pp. 199–256.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Leonid B. Golinskiĭ)

Referenced by:

§18.32.

Encodings:

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On the function $H(m,a,x)=\exp (-ix)F(m+1-ia,2m+2;2ix)$ . J. Math. Phys. Mass. Inst. Tech. 21, pp. 264–283.

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External Links:

ISSN 0097-1421, MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§33.7.

Encodings:

BibTeX

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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8: 24.2 Definitions and Generating Functions

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$B_{2n+1}=0$ ,

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${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

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$E_{2n+1}=0$ ,

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${(-1)}^n{E}_{2n}>0$ .

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24.2.9 $$E_n=2^n{E}_n\left(\frac{1}{2}\right)=\text{integer},$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.2

Permalink:

http://dlmf.nist.gov/24.2.E9

Encodings:

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See also:

Annotations for §24.2(ii), §24.2 and Ch.24

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9: 29 Lamé Functions

Chapter 29 Lamé Functions

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10: Bibliography G

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

BibTeX

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Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

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Notes:

For certification see same journal 8(1965), pp. 105–106, for remark see ACM Trans. Math. Software, 1(1975), pp. 282–284. Includes Algol program, maximum accuracy: 11D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(iii).

Encodings:

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Handbook of Combinatorics. Vols. 1, 2. Elsevier Science B.V., Amsterdam.

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External Links:

ISBN 0-444-88002-X, MathReview, ZentralBlatt

Referenced by:

§26.19, §26.20, Ch.26.

Encodings:

BibTeX

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A table of partitions. Proc. London Math. Soc. (2) 39, pp. 142–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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A table of partitions (II). Proc. London Math. Soc. (2) 42, pp. 546–549.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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