Rogers%E2%80%93Ramanujan%20identities

SearchAdvancedHelp

(0.002 seconds)

21—30 of 249 matching pages

21: 10.23 Sums

… ►For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). …

22: 9.18 Tables

… ►

Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

… ►

Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

►

Yakovleva (1969) tabulates Fock’s functions $U(x)\equiv \sqrt{\pi }\mathrm{Bi}\left(x\right)$, $U^{\prime }(x)=\sqrt{\pi }{\mathrm{Bi}}^{\prime }\left(x\right)$, $V(x)\equiv \sqrt{\pi }\mathrm{Ai}\left(x\right)$, $V^{\prime }(x)=\sqrt{\pi }{\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-9(.001)9$. Precision is 7S.

… ►

Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

… ►

National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

…

23: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

24: Bibliography J

… ►

Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

ⓘ

External Links:

ISSN 0167-2789, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

… ►

Efficient implementation of the Hardy-Ramanujan-Rademacher formula. LMS J. Comput. Math. 15, pp. 341–359.

ⓘ

External Links:

ISSN 1461-1570, Document, Link, MathReview (Tim Huber), ZentralBlatt

Referenced by:

§27.20.

Encodings:

BibTeX

… ►

Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.

ⓘ

External Links:

ISSN 0305-0041, Document, MathReview (Cataldo Agostinelli)

Referenced by:

§34.12.

Encodings:

BibTeX

…

25: 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

… ►

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

…

26: 20.11 Generalizations and Analogs

… ► ►

§20.11(ii) Ramanujan’s Theta Function and $q$-Series

►Ramanujan’s theta function $f(a,b)$ is defined by … ►

§20.11(iii) Ramanujan’s Change of Base

… ►These results are called Ramanujan’s changes of base. …

27: Bibliography P

… ►

Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(iv), §18.33(v).

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

… ►

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

…

28: Bibliography M

… ►

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

… ►

Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

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

…

29: 27.14 Unrestricted Partitions

… ►

§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity …Ramanujan also found that $p\left(7n+5\right)\equiv 0(mod7)$ and $p\left(11n+6\right)\equiv 0(mod11)$ for all $n$. … ►

30: 18.28 Askey–Wilson Class

… ►These polynomials are also called Rogers polynomials. …