Bailey%20pairs
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1: 17.12 Bailey Pairs
§17.12 Bailey Pairs
… ►Bailey Pairs
… ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: … ►The Bailey pair and Bailey chain concepts have been extended considerably. …2: 17 q-Hypergeometric and Related Functions
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3: 27.15 Chinese Remainder Theorem
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►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
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►Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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4: Bibliography B
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A Fortran-90 based multiprecision system.
ACM Trans. Math. Software 21 (4), pp. 379–387.
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Products of generalized hypergeometric series.
Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
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Transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
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The generating function of Jacobi polynomials.
J. London Math. Soc. 13, pp. 8–12.
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Generalized Hypergeometric Series.
Stechert-Hafner, Inc., New York.
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5: 4.48 Software
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►See also Bailey (1995), Hull and Abrham (1986), Xu and Li (1994).
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Bailey (1993). Fortran.
6: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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A trinomial analogue of Bailey’s lemma and superconformal invariance.
Comm. Math. Phys. 192 (2), pp. 245–260.
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Umbral calculus, Bailey chains, and pentagonal number theorems.
J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
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Bailey’s Transform, Lemma, Chains and Tree.
In Special Functions 2000: Current Perspective and Future
Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.),
NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.
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7: 20 Theta Functions
Chapter 20 Theta Functions
…8: 28.17 Stability as
§28.17 Stability as
►If all solutions of (28.2.1) are bounded when along the real axis, then the corresponding pair of parameters is called stable. All other pairs are unstable. ►For example, positive real values of with comprise stable pairs, as do values of and that correspond to real, but noninteger, values of . ►However, if , then always comprises an unstable pair. …9: 10.75 Tables
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Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of and for , 8D.
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
Zhang and Jin (1996, p. 322) tabulates , , , , , , , , , 7S.