DLMF: Untitled Document

… ►To compute a particular value

p\left(n\right)

it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function

\tau\left(n\right)

, and the values can be checked by the congruence (27.14.20). …

… ►

Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from

A_{j,k}

.

► ►►►

	$p\left(\mathcal{D},n\right)$	$p\left(\mathcal{D}2,n\right)$	$p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)$	$p\left(\mathcal{D}^{\prime}3,n\right)$
…
$20$	$64$	$31$	$20$	$18$

► … ►

§26.10(iv) Identities

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

26.10.16

p\left(\mathcal{D},n\right)\sim\frac{{\mathrm{e}}^{\pi\sqrt{n/3}}}{(768n^{3})^% {1/4}},

n\to\infty

.

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $p\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of partitions of $n$ and $n$ : nonnegative integer
A&S Ref:: 24.2.2
Permalink:: http://dlmf.nist.gov/26.10.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §26.10(v), §26.10 and Ch.26

… ►

26.10.18

A_{k}(n)=\sum_{\begin{subarray}{c}1<h\leq k\\ \left(h,k\right)=1\end{subarray}}{\mathrm{e}}^{\pi\mathrm{i}f(h,k)-(2\pi% \mathrm{i}nh/k)},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $\mathrm{i}$ : imaginary unit, $h$ : nonnegative integer, $k$ : nonnegative integer, $n$ : nonnegative integer, $A_{k}(n)$ : coefficient and $f(h,k)$ : function
Permalink:: http://dlmf.nist.gov/26.10.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §26.10(vi), §26.10 and Ch.26

…

… ►

P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

D. H. Lehmer (1943) Ramanujan’s function

\tau(n)

. Duke Math. J. 10 (3), pp. 483–492.

ⓘ

External Links:: Document, MathReview (H. S. Zuckerman), ZentralBlatt
Referenced by:: §27.20, §27.20, §27.21.
Encodings:: BibTeX

►

D. H. Lehmer (1947) The vanishing of Ramanujan’s function

\tau(n)

. Duke Math. J. 14 (2), pp. 429–433.

ⓘ

External Links:: Document, MathReview (R. A. Rankin), ZentralBlatt
Referenced by:: §27.14(vi).
Encodings:: BibTeX

… ►

J. Lepowsky and R. L. Wilson (1982) A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.16.
Encodings:: BibTeX

…

… ►Much has changed in the years since A&S was published. …However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. … ►November 20, 2009 …

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) 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

… ►

G. E. Andrews, R. A. Askey, B. C. Berndt, and R. A. Rankin (Eds.) (1988) 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

… ►

G. E. Andrews (1984) 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

…

… ►He is managing editor of the Ramanujan Journal, a journal devoted to areas of mathematics influenced by Ramanujan. …

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

… ►

C. de la Vallée Poussin (1896a) Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction

\zeta(s)

de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

ⓘ

Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

►

C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire

Mx+N

. Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

ⓘ

Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

… ►

H. Ding, K. I. Gross, and D. St. P. Richards (1996) 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

… ►

B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:: (This reference has several typographical errors.)
External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.21(ii), §13.21(iii), §13.21(iv).
Encodings:: BibTeX

…

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

Ramanujan%20change%20of%20base

11—20 of 491 matching pages

11: 27.20 Methods of Computation: Other Number-Theoretic Functions

12: 26.10 Integer Partitions: Other Restrictions

§26.10(iv) Identities

13: Bibliography L

14: Foreword

15: Bibliography

16: Frank Garvan

17: 8.26 Tables

18: Bibliography D

19: 36 Integrals with Coalescing Saddles

20: Gergő Nemes