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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: 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. …
3: 24.19 Methods of Computation
For number-theoretic applications it is important to compute B 2 n ( mod p ) for 2 n p - 3 ; in particular to find the irregular pairs ( 2 n , p ) for which B 2 n 0 ( mod p ) . We list here three methods, arranged in increasing order of efficiency. …
  • A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs ( 2 n , p ) . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

  • 4: 32.4 Isomonodromy Problems
    P I P VI  can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose …
    5: 10.25 Definitions
    §10.25(iii) Numerically Satisfactory Pairs of Solutions
    Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). …
    Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.
    Pair Region
    6: 27.5 Inversion Formulas
    27.5.3 g ( n ) = d | n f ( d ) f ( n ) = d | n g ( d ) μ ( n d ) .
    Special cases of Möbius inversion pairs are: …
    7: 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. …
    8: 27.19 Methods of Computation: Factorization
    Type II probabilistic algorithms for factoring n rely on finding a pseudo-random pair of integers ( x , y ) that satisfy x 2 y 2 ( mod n ) . …
    9: 9.2 Differential Equation
    §9.2(iii) Numerically Satisfactory Pairs of Solutions
    Table 9.2.1 lists numerically satisfactory pairs of solutions of (9.2.1) for the stated intervals or regions; compare §2.7(iv).
    Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.
    Pair Interval or Region
    10: 18.38 Mathematical Applications
    Integrable Systems
    It has elegant structures, including N -soliton solutions, Lax pairs, and Bäcklund transformations. …