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1: 20.11 Generalizations and Analogs
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§20.11(ii) Ramanujan’s Theta Function and -Series
… ►In the case identities for theta functions become identities in the complex variable , with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As in §20.11(ii), the modulus of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in -series via (20.9.1). However, in this case is no longer regarded as an independent complex variable within the unit circle, because is related to the variable of the theta functions via (20.9.2). … ►For applications to rapidly convergent expansions for see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). …2: Bibliography K
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Asymptotic expansions of certain -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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Theorie und Anwendung der unendlichen Reihen.
4th edition, Die Grundlehren der mathematischen Wissenschaften, Band 2, Springer-Verlag, Berlin-Heidelberg (German).
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Electromagnetic Fields and Relativistic Particles.
International Series in Pure and Applied Physics, McGraw-Hill Book Co., New York.
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Dualities in the -Askey scheme and degenerate DAHA.
Stud. Appl. Math. 141 (4), pp. 424–473.
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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively -binomial sums and basic hypergeometric series.
Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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3: Bibliography S
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Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity.
4th edition, Springer-Verlag, Berlin.
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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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Multiplicative processes and power laws.
Phys. Rev. E 57 (4), pp. 4811–4813.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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An Introduction to Basic Fourier Series.
Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
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4: Bibliography G
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Basic Hypergeometric Series.
Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Matrices, moments and quadrature with applications.
Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.
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Numerical Analysis of Spectral Methods: Theory and Applications.
Society for Industrial and Applied Mathematics, Philadelphia, PA.
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Multilateral summation theorems for ordinary and basic hypergeometric series in
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SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
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