DLMF: Untitled Document

… ►

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.

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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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S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

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G. Almkvist and B. Berndt (1988) 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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G. E. Andrews, R. A. Askey, B. C. Berndt, and R. A. Rankin (Eds.) (1988) 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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G. E. Andrews (1984) 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

…

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

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A. J. Jerri (1982) A note on sampling expansion for a transform with parabolic cylinder kernel. Inform. Sci. 26 (2), pp. 155–158.

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External Links:: ISSN 0020-0255, Document, MathReview, ZentralBlatt
Referenced by:: §12.16.
Encodings:: BibTeX

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H. K. Johansen and K. Sørensen (1979) Fast Hankel transforms. Geophysical Prospecting 27 (4), pp. 876–901.

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External Links:: Document
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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F. Johansson (2012) 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.
Encodings:: BibTeX

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G. S. Joyce (1973) On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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G. S. Joyce (1994) On the cubic lattice Green functions. Proc. Roy. Soc. London Ser. A 445, pp. 463–477.

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External Links:: ISSN 0962-8444, FullText, MathReview (Peter J. Forrester), ZentralBlatt
Referenced by:: §31.17(ii), §31.7(i).
Encodings:: BibTeX

…

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M. Katsurada (2003) 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).
Encodings:: BibTeX

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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T. H. Koornwinder (2009) 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).
Encodings:: BibTeX

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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. …

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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.

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

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

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

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

\tau(n)

. Duke Math. J. 10 (3), pp. 483–492.

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

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

\tau(n)

. Duke Math. J. 14 (2), pp. 429–433.

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

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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.

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

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Bailey Transform

… ► ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …

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R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

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P. I. Pastro (1985) 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).
Encodings:: BibTeX

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E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

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R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

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A. Pinkus and S. Zafrany (1997) 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).
Encodings:: BibTeX

…

… ►

Cubic Equations

… ►For the roots

\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}

of

g(w)=0

and the roots

\theta_{1},\theta_{2},\theta_{3}

of the resolvent cubic equation … ►

1.11.20

\sqrt{-\theta_{1}}\;\sqrt{-\theta_{2}}\;\sqrt{-\theta_{3}}=-q.

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Symbols:: $q$ and $\theta_{j}$ : cubic roots
Permalink:: http://dlmf.nist.gov/1.11.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §1.11(iii), §1.11(iii), §1.11 and Ch.1

… ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

,

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

,

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

,

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

,

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. …

… ►

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).

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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.

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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.

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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

…

Ramanujan%20cubic%20transformation

11—20 of 267 matching pages

11: Bibliography

12: Frank Garvan

13: Bibliography J

14: Bibliography K

15: 26.10 Integer Partitions: Other Restrictions

§26.10(iv) Identities

16: Bibliography L

17: 17.12 Bailey Pairs

Bailey Transform

18: Bibliography P

19: 1.11 Zeros of Polynomials

Cubic Equations

20: Bibliography D