Ramanujan%20tau%20function

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1: 20.11 Generalizations and Analogs

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§20.11(ii) Ramanujan’s Theta Function and $q$-Series

►Ramanujan’s theta function $f(a,b)$ is defined by … ►

§20.11(iii) Ramanujan’s Change of Base

… ►However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). … ►These results are called Ramanujan’s changes of base. …

2: Bibliography L

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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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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

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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.

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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.

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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).

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

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.16.

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3: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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4: Bibliography W

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Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.

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ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§22.20(ii), §22.20(v).

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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External Links:

Document

Referenced by:

(20.7.34), §20.7(ix).

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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

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External Links:

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

Referenced by:

§27.21.

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5: Bibliography K

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

Referenced by:

§17.2(i).

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

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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).

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A Program for Computing the Conical Functions of the First Kind $P_{-1/2+i\tau }^m(x)$ for $m=0$ and $m=1$ . Comput. Phys. Comm. 23 (1), pp. 51–61.

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Notes:

Single-precision Fortran, maximum accuracy 15D.

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Document, Code

Referenced by:

§14.31(ii), 1st item.

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

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

Referenced by:

§18.26(v), §18.26(v).

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