Bailey pairs

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11: 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 $x^2\equiv y^2(modn)$. …

12: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Bailey’s Bilateral Summations

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13: 18.18 Sums

… ►and a similar pair of equations by symmetry; compare the second row in Table 18.6.1. … ►For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938). …

14: 9.2 Differential Equation

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§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.

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Pair	Interval or Region
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15: 16.6 Transformations of Variable

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16: 17.1 Special Notation

… ►A slightly different notation is that in Bailey (1964) and Slater (1966); see §17.4(i). …

17: 17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

§17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

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Bailey’s ${}_2{}^{}\psi _2^{}$ Transformations

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18: 13.28 Physical Applications

… ►and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …

19: 15.17 Mathematical Applications

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. …

20: Bille C. Carlson

… ►In his paper Lauricella’s hypergeometric function $F_D$ (1963), he defined the $R$-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …