Ramanujan 1ψ1 summation

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11: Bibliography R

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Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

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External Links:

ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:

§24.7(i).

Encodings:

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Collected Papers of Srinivasa Ramanujan. Chelsea Publishing Co., New York.

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Notes:

Reprinted with commentary by Bruce Berndt, AMS Chelsea Publishing Series, American Mathematical Society (2000).

External Links:

ISBN 0-8218-2076-1

Referenced by:

§8.11(v), G. H. Hardy and S. Ramanujan (1918).

Encodings:

BibTeX

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Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis. Elsevier, New York.

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Notes:

revised and expanded Edition from 1972,

External Links:

ISBN 10: 0-12-585050-6(vol. 1), MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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12: 17.12 Bailey Pairs

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ … ► ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: ►

${\alpha }_n={\displaystyle \frac{{(a;q)}_n(1-a{q}^{2n}){(-1)}^n{q}^{n(3n-1)/2}{a}^n}{{(q;q)}_n(1-a)}},$

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${\beta }_n={\displaystyle \frac{1}{{(q;q)}_n}}.$

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13: 5.13 Integrals

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5.13.1 $$\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(s+a\right)\mathrm{\Gamma }\left(b-s\right)z^{-s}ds=\frac{\mathrm{\Gamma }\left(a+b\right)z^a}{{(1+z)}^{a+b}},$$ $\mathrm{\Re }\left(a+b\right)>0$, , .

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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, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $s$: real or complex variable, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13, §5.13, §5.19(ii)

Permalink:

http://dlmf.nist.gov/5.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13 and Ch.5

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5.13.2 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|\mathrm{\Gamma }\left(a+\mathrm{i}t\right)|}^2{\mathrm{e}}^{(2b-\pi )t}dt=\frac{\mathrm{\Gamma }\left(2a\right)}{{(2\sin b)}^{2a}},$$ $a>0$, .

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E2

Encodings:

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See also:

Annotations for §5.13 and Ch.5

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Ramanujan’s Beta Integral

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5.13.4 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dt}{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(b+t\right)\mathrm{\Gamma }\left(c-t\right)\mathrm{\Gamma }\left(d-t\right)}=\frac{\mathrm{\Gamma }\left(a+b+c+d-3\right)}{\mathrm{\Gamma }\left(a+c-1\right)\mathrm{\Gamma }\left(a+d-1\right)\mathrm{\Gamma }\left(b+c-1\right)\mathrm{\Gamma }\left(b+d-1\right)},$$ $\mathrm{\Re }\left(a+b+c+d\right)>3$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

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5.13.5 $\mathrm{\Re }\left(a_k\right)>0$, $k=1,2,3,4$.

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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, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

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14: 17.13 Integrals

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17.13.1 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-ax/c;q)}_{\mathrm{\infty }}{(bx/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{(1-q){(q;q)}_{\mathrm{\infty }}{(ab;q)}_{\mathrm{\infty }}cd{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(a;q)}_{\mathrm{\infty }}{(b;q)}_{\mathrm{\infty }}(c+d){(-bc/d;q)}_{\mathrm{\infty }}{(-ad/c;q)}_{\mathrm{\infty }}},$$

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Symbols:

$\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E1

Encodings:

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See also:

Annotations for §17.13 and Ch.17

►or, when , … ►

Ramanujan’s Integrals

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

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Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

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17.13.4 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-ct{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-ct;q)}_{\mathrm{\infty }}}{d}_{q}t=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right){(-c{q}^{\alpha };q)}_{\mathrm{\infty }}{(-q^{1-\alpha }/c;q)}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right){(-c;q)}_{\mathrm{\infty }}{(-q/c;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13, §17.13 and Ch.17

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15: 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:

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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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16: Richard A. Askey

… ► Schempp), published by Reidel Publishing Company in 1984, Ramanujan Revisited (with G. … ►

1 Algebraic and Analytic Methods

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17: Bibliography

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Chapter $16$ of Ramanujan’s second notebook: Theta-functions and $q$-series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.

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External Links:

ISSN 0065-9266, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§20.11(ii).

Encodings:

BibTeX

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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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Ramanujan Revisited. Academic Press Inc., Boston, MA.

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External Links:

ISBN 0-12-058560-X, ISBN 0-12-058560-X, MathReview, ZentralBlatt

Referenced by:

§20.11(ii), Profile Richard A. Askey.

Encodings:

BibTeX

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Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:

MathReview (R. N. Jain), ZentralBlatt

Referenced by:

§17.11.

Encodings:

BibTeX

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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18: 17.14 Constant Term Identities

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17.14.1

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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Rogers–Ramanujan Constant Term Identities

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17.14.2 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}q;q)}_{\mathrm{\infty }}}=\frac{H(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $H(q)$: LHS of (17.2.50)

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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17.14.3 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q)}_{\mathrm{\infty }}}=\frac{G(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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17.14.4 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2}}{{(q^2;q^2)}_n{(q;q^2)}_n}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-z^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1};q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-2};q^4)}_{\mathrm{\infty }}}=\frac{G(q^4)}{{(q;q^2)}_{\mathrm{\infty }}},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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19: David M. Bressoud

… ►His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …

20: Peter Paule

… ►He is on the editorial boards for the Journal of Symbolic Computation and The Ramanujan Journal, and is Managing Editor of Annals of Combinatorics. …