Ramanujan 1ψ1 summation

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21: Wadim Zudilin

… ►He received the Distinguished Award of the Hardy–Ramanujan Society in 2001 and was one of the co-recipients of the 2014 G. …

22: 26.10 Integer Partitions: Other Restrictions

… ►Throughout this subsection it is assumed that . … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. … ►

§26.10(iv) Identities

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

23: Bibliography H

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Tables of Radial Spheroidal Wave Functions, Vols. 1-3, Prolate, $m=0,1,2$; Vols. 4-6, Oblate, $m=0,1,2$ . Technical report Naval Research Laboratory, Washington, D.C..

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NRL Reports 7088-7093

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ZentralBlatt

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3rd item.

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Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, pp. 75–115.

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Reprinted in Ramanujan (1962, pp. 276–309).

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Document, MathReview Entry, ZentralBlatt

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§27.14(iii).

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Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press, Cambridge, England.

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Reprinted by Chelsea Publishing Company, New York, 1959.

External Links:

MathReview (H. Rademacher), ZentralBlatt

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§27.13(iv).

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

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§24.16(iii).

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25D Table of the First One Hundred Values of $j_{0,s}$,$J_1(j_{0,s})$, $j_{1,s}$,$J_0(j_{1,s})=J_0(j_{0,s+1}^{\prime })$,$j_{1,s}^{\prime }$,$J_1(j_{1,s}^{\prime })$ . Technical report Department of Physics, Worcester Polytechnic Institute, Worcester, MA.

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Ms. of six pages deposited in the UMT file

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8th item.

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24: Bibliography W

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Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.

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ISSN 0006-3835, Document, MathReview Entry, ZentralBlatt

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§3.5(iv).

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Generating functions of class-numbers. Compositio Math. 1, pp. 39–68.

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ISSN 0010-437X, FullText, MathReview Entry, ZentralBlatt

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§20.7(v), §20.8.

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

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§27.21.

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

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Document

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§2.11(vi), §3.9(v), §3.9(vi).

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Rational Chebyshev approximations for the Bessel functions $J_0(x)$, $J_1(x)$, $Y_0(x)$, $Y_1(x)$ . Math. Comp. 39 (160), pp. 617–623.

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Tables on microfiche

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

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9th item.

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25: 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)). …

26: Bibliography M

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§16.10.

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

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§17.15.

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

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§17.15.

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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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Document, ZentralBlatt

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§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

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§27.13(iv).

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27: 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

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§17.2(i).

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An extension of Saalschütz’s summation theorem for the series ${}_{r+3}{}^{}F_{r+2}^{}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt

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§16.4(ii), §16.4(ii).

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Orthogonal polynomials with weight function ${(1-x)}^{\alpha }{(1+x)}^{\beta }+M\delta (x+1)+N\delta (x-1)$ . Canad. Math. Bull. 27 (2), pp. 205–214.

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ISSN 0008-4395, MathReview (Wolfgang Hahn), ZentralBlatt

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§18.36(i), §18.36(i).

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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

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§18.26(v), §18.26(v).

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Some special cases of the generalized hypergeometric function ${}_{q+1}{}^{}F_q^{}$ . J. Comput. Appl. Math. 78 (1), pp. 79–95.

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ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

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§16.7.

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28: Bibliography C

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Computing elliptic integrals by duplication. Numer. Math. 33 (1), pp. 1–16.

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ISSN 0029-599X, Document, MathReview, ZentralBlatt

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§19.19.

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On Ramanujan’s cubic transformation formula for ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};1;z)$ . Math. Proc. Cambridge Philos. Soc. 124 (2), pp. 193–204.

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ISSN 0305-0041, Document, MathReview (Aleksander Grytczuk), ZentralBlatt

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§15.8(v).

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

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§26.8(vii).

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Approximations and Complex Multiplication According to Ramanujan. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), G. E. Andrews, R. A. Askey, B. C. Bernd, K. G. Ramanathan, and R. A. Rankin (Eds.), pp. 375–472.

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External Links:

MathReview (Maurice Mignotte), ZentralBlatt

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§20.11(iii).

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A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.

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FullText, MathReview, ZentralBlatt

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§18.40(i).

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29: 19.9 Inequalities

… ►Throughout this subsection , except in (19.9.4). …for $0\le k\le 1$. … ►The earliest is due to Kepler and the most accurate to Ramanujan. Ramanujan’s approximation and its leading error term yield the following approximation to $L(a,b)/(\pi (a+b))$: …Barnard et al. (2000) shows that nine of the thirteen approximations, including Ramanujan’s, are from below and four are from above. …

30: Bibliography L

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Ramanujan’s function $\tau (n)$ . Duke Math. J. 10 (3), pp. 483–492.

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Document, MathReview (H. S. Zuckerman), ZentralBlatt

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§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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Document, MathReview (R. A. Rankin), ZentralBlatt

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§27.14(vi).

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List of Prime Numbers from 1 to 10,006,721. Publ. No. 165, Carnegie Institution of Washington, Washington, D.C..

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ZentralBlatt

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§27.21.

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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

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§17.16.

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

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