Ramanujan’s

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21: 19.35 Other Applications

… ►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …

22: Peter Paule

… ►Paule’s main research interests are computer algebra and algorithmic mathematics, together with connections to combinatorics, special functions, number theory, and other related fields. He is on the editorial boards for the Journal of Symbolic Computation and The Ramanujan Journal, and is Managing Editor of Annals of Combinatorics. … ►Paule was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …

23: 26.10 Integer Partitions: Other Restrictions

… ► $p(\in S,n)$ denotes the number of partitions of $n$ into parts taken from the set $S$. … ►

26.10.5 $$\sum _{n=0}^{\mathrm{\infty }}p(\in S,n)q^n=\prod _{j\in S}\frac{1}{1-q^j}.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $n$: nonnegative integer, $S$: set and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.10.E5

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See also:

Annotations for §26.10(ii), §26.10 and Ch.26

… ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. ►

§26.10(iv) Identities

►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …

24: 27.21 Tables

… ►7 of Abramowitz and Stegun (1964) also lists the factorizations in Glaisher’s Table I(a); Table 24. … ►Tables of the Ramanujan function $\tau \left(n\right)$ are published in Lehmer (1943) and Watson (1949). …The bibliography in Lehmer (1941) gives references to the places in Dickson’s History where the older tables are cited. …

25: Bibliography P

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The asymptotic behaviour of Pearcey’s integral for complex variables. Proc. Roy. Soc. London Ser. A 432 (1886), pp. 391–426.

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External Links:

ISSN 0962-8444, Document, Link, MathReview (J. S. Joel), ZentralBlatt

Referenced by:

§36.12(i).

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

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External Links:

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

Referenced by:

§18.33(iv), §18.33(v).

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

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Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

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Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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Referenced by:

2nd item.

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26: 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.  227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S.  Wagon), published by Key College Press in 2000, and A Radical Approach to Lebesgue’s Theory of Integration, published by the Mathematical Association of America and Cambridge University Press in 2007. …

27: Bibliography J

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

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

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External Links:

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

Referenced by:

§27.20.

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

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Asymptotic behavior of integrals. SIAM Rev. 14 (2), pp. 286–317.

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External Links:

Document, MathReview (A. Erdelyi), ZentralBlatt

Referenced by:

Ch.2.

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On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§32.12(i).

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28: 8.11 Asymptotic Approximations and Expansions

… ►For Dawson’s integral $F\left(y\right)$ see §7.2(ii). … ►If $S_n(x)$ is defined by ►

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

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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

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See also:

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

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

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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

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See also:

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

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29: Bibliography K

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A proof of Askey’s conjectured $q$-analogue of Selberg’s integral and a conjecture of Morris. SIAM J. Math. Anal. 19 (4), pp. 969–986.

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External Links:

ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.13.

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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.

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External Links:

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

Referenced by:

§17.2(i).

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Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17 (5), pp. 1232–1258.

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External Links:

ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§25.12(ii).

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

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External Links:

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

Referenced by:

§18.26(v), §18.26(v).

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Tom Koornwinder’s Personal Collection of Maple Procedures

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External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

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30: Richard A. Askey

… ► Schempp), published by Reidel Publishing Company in 1984, Ramanujan Revisited (with G. … ►S. … ►Askey was a member of the original editorial committee for the DLMF project, serving as an Associate Editor advising on all aspects of the project from the mid-1990’s to the mid-2010’s when the organizational structure of the DLMF project was reconstituted; see About the Project.