Bailey pairs

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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: 28.17 Stability as $x\to\pm\infty$

§28.17 Stability as $x\to\pm\infty$

►If all solutions of (28.2.1) are bounded when

x\to\pm\infty

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

\nu

. ►However, if

\Im\nu\neq 0

, then

(a,q)

always comprises an unstable pair. …

4: 4.48 Software

… ►

•

Bailey (1993). Fortran.

… ►See also Bailey (1995), Hull and Abrham (1986), Xu and Li (1994). …

5: 24.19 Methods of Computation

… ►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{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 $(2n,p)$ . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

6: 32.4 Isomonodromy Problems

… ►

\mbox{P}_{\mbox{\scriptsize I}}

–

\mbox{P}_{\mbox{\scriptsize VI}}

can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose …

7: 5.24 Software

… ►

•

Bailey (1993). Fortran and C++ wrapper.

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8: 10.25 Definitions

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§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
…

►

9: 27.5 Inversion Formulas

… ►

27.5.3

g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mu\left(\frac{n}{d}\right).

ⓘ

Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer, $n$ : positive integer, $f$ : function and $g$ : function
A&S Ref:: 24.3.1 II.C
Permalink:: http://dlmf.nist.gov/27.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

►Special cases of Möbius inversion pairs are: …

10: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{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. …