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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
3: 27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod m k ) , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod m ), where m is the product of the moduli. … Their product m 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 ( mod m ) , which is correct to 20 digits. …
4: Bibliography B
  • D. H. Bailey (1995) A Fortran-90 based multiprecision system. ACM Trans. Math. Software 21 (4), pp. 379–387.
  • W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
  • W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
  • W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.
  • W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.
  • 5: 4.48 Software
  • Bailey (1993). Fortran.

  • See also Bailey (1995), Hull and Abrham (1986), Xu and Li (1994). …
    6: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • G. E. Andrews and A. Berkovich (1998) A trinomial analogue of Bailey’s lemma and N = 2 superconformal invariance. Comm. Math. Phys. 192 (2), pp. 245–260.
  • G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
  • G. E. Andrews (2001) 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.
  • 7: 20 Theta Functions
    Chapter 20 Theta Functions
    8: 28.17 Stability as x ±
    §28.17 Stability as x ±
    If all solutions of (28.2.1) are bounded when x ± along the real axis, then the corresponding pair of parameters ( a , q ) is called stable. All other pairs are unstable. For example, positive real values of a with q = 0 comprise stable pairs, as do values of a and q that correspond to real, but noninteger, values of ν . However, if ν 0 , then ( a , q ) always comprises an unstable pair. …
    9: 10.75 Tables
  • Achenbach (1986) tabulates J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) , x = 0 ( .1 ) 8 , 20D or 18–20S.

  • Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of Y n ( z ) and Y n ( z ) for n = 0 ( 1 ) 5 , 8D.

  • Bickley et al. (1952) tabulates x n I n ( x ) or e x I n ( x ) , x n K n ( x ) or e x K n ( x ) , n = 2 ( 1 ) 20 , x = 0 (.01 or .1) 10(.1) 20, 8S; I n ( x ) , K n ( x ) , n = 0 ( 1 ) 20 , x = 0 or 0.1 ( .1 ) 20 , 10S.

  • Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of K n ( z ) and K n ( z ) , for n = 2 ( 1 ) 20 , 9S.

  • Zhang and Jin (1996, p. 322) tabulates ber x , ber x , bei x , bei x , ker x , ker x , kei x , kei x , x = 0 ( 1 ) 20 , 7S.

  • 10: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • S. O. Warnaar (1998) A note on the trinomial analogue of Bailey’s lemma. J. Combin. Theory Ser. A 81 (1), pp. 114–118.