# Bailey pairs

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##### 1: 17.12 Bailey Pairs
###### BaileyPairs
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. …
##### 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.

• ##### 8: 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). …
##### 9: 27.5 Inversion Formulas
27.5.3 $g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mu\left(\frac{n}{d}\right).$
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. …