Ramanujan sum

SearchAdvancedHelp

(0.001 seconds)

11—20 of 21 matching pages

11: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

… ►

17.8.1 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-z)}^n{q}^{n(n-1)/2}={(q,z,q/z;q)}_{\mathrm{\infty }};$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

… ►

17.8.3 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{n(3n-1)/2}{z}^{3n}(1+z{q}^n)={(q,-z,-q/z;q)}_{\mathrm{\infty }}{(q{z}^2,q/z^2;q^2)}_{\mathrm{\infty }}.$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

Sum Related to (17.6.4)

…

12: Richard A. Askey

… ►Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. … ► Schempp), published by Reidel Publishing Company in 1984, Ramanujan Revisited (with G. …

13: 17.14 Constant Term Identities

… ►

Rogers–Ramanujan Constant Term Identities

… ►

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

ⓘ

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

►

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

ⓘ

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

►

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

ⓘ

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

►

17.14.5 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2+2n}}{{(q^2;q^2)}_n{(q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-q^2{z}^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(q^4{z}^{-2};q^4)}_{\mathrm{\infty }}}=\frac{H(q^4)}{{(q;q^2)}_{\mathrm{\infty }}}.$$

ⓘ

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.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

…

14: Bibliography D

… ►

Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

… ►

A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

ⓘ

External Links:

ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

Encodings:

BibTeX

… ►

Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

ⓘ

External Links:

ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

Encodings:

BibTeX

… ►

Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

ⓘ

Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

…

15: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. … ►Tables of the Ramanujan function $\tau \left(n\right)$ are published in Lehmer (1943) and Watson (1949). …

16: Bibliography K

… ►

The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

… ►

Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

ⓘ

External Links:

ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

Encodings:

BibTeX

… ►

The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

ⓘ

External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

… ►

Estimates of trigonometric sums and their applications. Uspehi Mat. Nauk 13 (4 (82)), pp. 185–192 (Russian).

ⓘ

External Links:

MathReview (R. A. Rankin), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

… ►

HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

ⓘ

Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

…

17: Bibliography W

… ►

A table of Ramanujan’s function $\tau (n)$ . Proc. London Math. Soc. (2) 51, pp. 1–13.

ⓘ

External Links:

Document, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

… ►

Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

ⓘ

External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

…

18: Bibliography J

… ►

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

… ►

Note sur la série ${\sum }_{n=1}^{\mathrm{\infty }}{x}^n/n^s$ . Bull. Soc. Math. France 17, pp. 142–152 (French).

ⓘ

External Links:

ISSN 0037-9484, MathReview Entry, ZentralBlatt

Referenced by:

§25.12(ii).

Encodings:

BibTeX

…

19: 8.11 Asymptotic Approximations and Expansions

… ►

8.11.2 $$\mathrm{\Gamma }(a,z)=z^{a-1}{\mathrm{e}}^{-z}\left(\sum _{k=0}^{n-1}\frac{u_k}{z^k}+R_n(a,z)\right),$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $n$: nonnegative integer, $u_k$ and $R_n(a,z)$: remainder

A&S Ref:

6.5.32

Permalink:

http://dlmf.nist.gov/8.11.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

8.11.4 $$\gamma (a,z)=z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{{\left(a\right)}_{k+1}},$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

A&S Ref:

6.5.33 (corrected)

Referenced by:

§8.11(ii)

Permalink:

http://dlmf.nist.gov/8.11.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

8.11.6 $$\gamma (a,z)\sim -z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ , $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Notes:

See Paris (2002b)

Referenced by:

§8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, . It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where and $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

See also:

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

►

8.11.7 $$\mathrm{\Gamma }(a,z)\sim z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Referenced by:

§8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, $\lambda >1$. It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where $\lambda >1$ and $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

See also:

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

… ►

8.11.18 $$S_n(x)\sim \sum _{k=0}^{\mathrm{\infty }}{d}_k(x)n^{-k},$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $S_n(x)$: function and $d_k(x)$: coefficients

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E18

Encodings:

pMML, png, TeX

See also:

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

…

20: Bibliography

… ►

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

ⓘ

External Links:

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

Referenced by:

§20.11(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§19.9(i).

Encodings:

BibTeX

… ►

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

ⓘ

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

… ►

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

ⓘ

External Links:

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

Referenced by:

§17.12.

Encodings:

BibTeX

… ►

Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

ⓘ

Referenced by:

§24.4(iii).

Encodings:

BibTeX

…