Ramanujan 1?1 summation

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1: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

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Bailey’s Bilateral Summations

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2: Bibliography B

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Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. The Macmillan Co., New York.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§11.4(i), §11.4(v), §11.5(ii), §11.5(i), §11.5(ii), §11.5(iii), §11.7(ii), §11.7(iii), §11.7(v), §11.9(i), §11.9(iv), §11.9(iv), Ch.11, §14.29.

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On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (4), pp. 1119–1178.

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External Links:

ISSN 0894-0347, FullText, MathReview (David J. Aldous), ZentralBlatt

Referenced by:

§32.14.

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Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§11.7(v), §11.7(v).

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Real values of the $W$-function. ACM Trans. Math. Software 21 (2), pp. 161–171.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.45(iii).

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Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.

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External Links:

ISSN 0090-5364, Document, MathReview, ZentralBlatt

Referenced by:

§35.6(iv), §35.7(iv), §35.7(iv), §35.9.

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3: Bibliography M

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Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables. Ellis Horwood Ltd./John Wiley & Sons, Inc, Chichester/New York.

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Notes:

Edited by F. D. Gakhov, Translated from the Russian by L. W. Longdon

External Links:

ISBN 0-85312-538-7, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.32(iv), §10.43(vi), §10.9(iv), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §6.14(iii), §7.14(iii), §8.14.

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On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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ISSN 1063-5203, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(iii).

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Über die asymptotische Entwicklung von ${\displaystyle {\int }_0^{-\mathrm{\infty }-i(\arg w-\mu )}}{e}^{\nu z-w\sinh z}𝑑z$, für große Werte von $|w|$ und $|\nu |$. I, II. Proc. Akad. Wet. Amsterdam 35, pp. 1170–1180, 1291–1303 (German).

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External Links:

ISSN 0370-0348, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

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On the representation of numbers as a sum of $2r$ squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:

ZentralBlatt

Referenced by:

§27.13(iv).

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BibTeX

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4: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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Table of Definite and Infinite Integrals. Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.

External Links:

ISBN 0-444-42151-3, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§1.4(iv), §10.22(vi), §10.43(vi), §11.7(v), §11.9(iv), §12.5(iv), §13.10(vi), §13.23(v), §15.14, §16.20, §16.5, §18.17(ix), §4.10(iii), §4.26(v), §4.40(v), §5.13, §6.14(iii), §7.14(iii), §8.14, §8.19(x).

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Derivatives and integrals with respect to the order of the Struve functions ${𝐇}_{\nu }(x)$ and ${𝐋}_{\nu }(x)$ . J. Math. Anal. Appl. 137 (1), pp. 17–36.

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External Links:

ISSN 0022-247X, Document, MathReview (A. McD. Mercer), ZentralBlatt

Referenced by:

§11.4(vi), §11.7(iv).

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Perturbation solutions of the ellipsoidal wave equation. Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.

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ISSN 0033-5606, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§29.11.

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Multichannel Rydberg spectroscopy of complex atoms. Reviews of Modern Physics 68, pp. 1015–1123.

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External Links:

Document

Referenced by:

§33.22(ii).

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5: Bibliography W

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A note on the trinomial analogue of Bailey’s lemma. J. Combin. Theory Ser. A 81 (1), pp. 114–118.

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External Links:

ISSN 0097-3165, Document, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.12.

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

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A Treatise on the Theory of Bessel Functions. 2nd edition, Cambridge University Press, Cambridge, England.

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Notes:

Reprinted in 1995.

External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§1.13(v), §10.1, §10.10, §10.12, §10.13, §10.14, §10.14, §10.15, §10.16, §10.17(iii), §10.18(iii), §10.19(ii), §10.19(iii), §10.2(ii), §10.21(i), §10.21(i), §10.21(ii), §10.21(iii), §10.21(iv), §10.21(vi), §10.21(xiii), (10.22.72), (10.22.78), §10.22(i), §10.22(ii), §10.22(iv), §10.22(vi), §10.23(iii), §10.23(iii), §10.23(iii), §10.23(i), §10.23(ii), §10.23(iii), §10.27, §10.29(i), §10.29(ii), §10.31, §10.32(i), §10.32(ii), §10.32(iii), §10.32(iv), §10.34, §10.4, §10.40(i), §10.40(ii), §10.42, §10.42, §10.43(iv), §10.43(vi), §10.44(ii), §10.54, §10.6(i), §10.6(ii), §10.60(iii), §10.60(iv), §10.8, (10.9.26), §10.9(i), §10.9(ii), §10.9(iii), §10.9(iv), Ch.10, §11.10(i), §11.10(ii), §11.10(iii), §11.10(ix), §11.10(v), §11.10(vi), (11.11.10), (11.11.18), (11.11.19), (11.11.8), §11.11(i), §11.11(iii), §11.2(i), §11.2(ii), §11.4(ii), §11.4(v), §11.5(i), §11.5(iii), §11.6(i), §11.6(iii), §11.6(iii), §11.6(iii), §11.7(ii), §11.9(i), §11.9(iii), §11.9(iv), Ch.11, §18.17(v), §2.5(i), Erratum (V1.0.21) for Equation (10.22.72).

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Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.5(ii).

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On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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External Links:

ISSN 1044-677X, FullText

Referenced by:

§28.4(vii).

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6: Bibliography C

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Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

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External Links:

ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt

Referenced by:

§18.28(iv).

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A dispersion analysis for difference schemes: Tables of generalized Airy functions. Math. Comp. 32 (144), pp. 1163–1170.

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External Links:

ISSN 0025-5718, FullText, MathReview (Philip Brenner), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

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A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: Application to Pearcey’s integral and its derivatives. J. Phys. A 15 (4), pp. 1179–1190.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§36.15(iii), §36.8.

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On the asymptotic expansion of Airy’s integral. Proc. Glasgow Math. Assoc. 6, pp. 113–115.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

(9.5.7), §9.5(ii).

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Chazy’s second-degree Painlevé equations. J. Phys. A 39 (39), pp. 11955–11971.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§32.7(vii).

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