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21: Bibliography Y
  • Z. M. Yan (1992) Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.
  • 22: Bibliography
  • L. V. Ahlfors (1966) Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.
  • 23: 23.21 Physical Applications
    Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots (§23.3(i)), and setting …
    23.21.5 ( ( v ) ( w ) ) ( ( w ) ( u ) ) ( ( u ) ( v ) ) 2 = ( ( w ) ( v ) ) 2 u 2 + ( ( u ) ( w ) ) 2 v 2 + ( ( v ) ( u ) ) 2 w 2 .
    24: 15.8 Transformations of Variable
    A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. …
    25: 18.2 General Orthogonal Polynomials
    It is assumed throughout this chapter that for each polynomial p n ( x ) that is orthogonal on an open interval ( a , b ) the variable x is confined to the closure of ( a , b ) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) …
    26: 31.17 Physical Applications
    §31.17(i) Addition of Three Quantum Spins
    We use vector notation [ 𝐬 , 𝐭 , 𝐮 ] (respective scalar ( s , t , u ) ) for any one of the three spin operators (respective spin values). Consider the following spectral problem on the sphere S 2 : 𝐱 2 = x s 2 + x t 2 + x u 2 = R 2 . …Introduce elliptic coordinates z 1 and z 2 on S 2 . … For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986). …
    27: 1.16 Distributions
    1.16.36 ( P ( 𝐃 ) u ) , ϕ = P ( u ) , ϕ = ( u ) , P ϕ ,
    1.16.37 ( P u ) , ϕ = P ( 𝐃 ) ( u ) , ϕ ,
    28: 15.19 Methods of Computation
    In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …
    29: Bibliography P
  • R. B. Paris (2001a) On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.
  • R. B. Paris (2001b) On the use of Hadamard expansions in hyperasymptotic evaluation. II. Complex variables. Proc. Roy. Soc. London Ser. A 457, pp. 2855–2869.
  • 30: 23.19 Interrelations
    23.19.3 J ( τ ) = g 2 3 g 2 3 27 g 3 2 ,