Ramanujan change of base

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1: 20.11 Generalizations and Analogs

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§20.11(iii) Ramanujan’s Change of Base

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

2: 27.14 Unrestricted Partitions

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§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

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3: 8.11 Asymptotic Approximations and Expansions

… ►If $x=a+{(2a)}^{\frac{1}{2}}y$ and $a\to +\mathrm{\infty }$, then ►

8.11.10 $$P(a+1,x)=\frac{1}{2}\mathrm{erfc}\left(-y\right)-\frac{1}{3}\sqrt{\frac{2}{\pi a}}(1+y^2){\mathrm{e}}^{-y^2}+O\left(a^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E10

Encodings:

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See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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8.11.11 $${\gamma }^{\ast }(1-a,-x)=x^{a-1}\left(-\cos \left(\pi a\right)+\frac{\sin \left(\pi a\right)}{\pi }\left(2\sqrt{\pi }F\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi }{a}}\left(1-y^2\right)\right){\mathrm{e}}^{y^2}+O\left(a^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E11

Encodings:

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See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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8.11.14 $${\mathrm{e}}^{nx}=e_n(nx)+\frac{{(nx)}^n}{n!}{S}_n(x),$$

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Defines:

$S_n(x)$: function (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer and $e_n(z)$: functions

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E14

Encodings:

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See also:

Annotations for §8.11(v), §8.11 and Ch.8

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8.11.15 $$S_n(x)=\frac{\gamma (n+1,nx)}{{(nx)}^n{\mathrm{e}}^{-nx}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $n$: nonnegative integer and $S_n(x)$: function

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E15

Encodings:

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See also:

Annotations for §8.11(v), §8.11 and Ch.8

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4: Bibliography B

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Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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External Links:

ISSN 0022-4715, Document, MathReview, ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.

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External Links:

ISSN 1431-0643, FullText, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:

ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§15.8(v), §20.11(iii).

Encodings:

BibTeX

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Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.

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External Links:

ISBN 0-387-96794-X, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§15.7.

Encodings:

BibTeX

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Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.

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External Links:

ISBN 0-387-97503-9, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§17.13.

Encodings:

BibTeX

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5: 15.8 Transformations of Variable

… ►With $\zeta ={\mathrm{e}}^{2\pi \mathrm{i}/3}(1-z)/\left(z-{\mathrm{e}}^{4\pi \mathrm{i}/3}\right)$ ►

15.8.32 $$\frac{{\left(1-z^3\right)}^a}{{\left(-z\right)}^{3a}}\left(\frac{1}{\mathrm{\Gamma }\left(a+\frac{2}{3}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}F(\genfrac{}{}{0pt}{}{a,a+\frac{1}{3}}{\frac{2}{3}};z^{-3})+\frac{{\mathrm{e}}^{\frac{1}{3}\pi \mathrm{i}}}{z\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(\frac{4}{3}\right)}F(\genfrac{}{}{0pt}{}{a+\frac{1}{3},a+\frac{2}{3}}{\frac{4}{3}};z^{-3})\right)=\frac{3^{\frac{3}{2}a+\frac{1}{2}}{\mathrm{e}}^{\frac{1}{2}a\pi \mathrm{i}}\mathrm{\Gamma }\left(a+\frac{1}{3}\right){(1-\zeta )}^a}{2\pi \mathrm{\Gamma }\left(2a+\frac{2}{3}\right){(-\zeta )}^{2a}}F(\genfrac{}{}{0pt}{}{a+\frac{1}{3},3a}{2a+\frac{2}{3}};{\zeta }^{-1}),$$ $|z|>1$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $a$: real or complex parameter and $\zeta$: change of variable

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/15.8.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(v), §15.8(v), §15.8 and Ch.15

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Ramanujan’s Cubic Transformation

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