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Andrews–Askey sum

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1: 17.6 ϕ 1 2 Function
AndrewsAskey Sum
2: 17.7 Special Cases of Higher ϕ s r Functions
Sum Related to (17.6.4)
3: Richard A. Askey
Profile
Richard A. Askey
Richard A. Askey (b. … Askey received his Ph. … Andrews and R. …  Andrews, B. …
4: Bibliography
  • G. E. Andrews, R. A. Askey, B. C. Berndt, and R. A. Rankin (Eds.) (1988) Ramanujan Revisited. Academic Press Inc., Boston, MA.
  • G. E. Andrews, R. Askey, and R. Roy (1999) Special Functions. Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.
  • R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.
  • R. Askey (1974) Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.
  • R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.
  • 5: 18.18 Sums
    §18.18 Sums
    §18.18(vi) Bateman-Type Sums
    Jacobi
    §18.18(viii) Other Sums
    6: 18.38 Mathematical Applications
    The Askey–Gasper inequality
    18.38.3 m = 0 n P m ( α , 0 ) ( x ) 0 , - 1 x 1 , α > - 1 , n = 0 , 1 , ,
    See, for example, Andrews et al. (1999, Chapter 9). …
    7: 14.30 Spherical and Spheroidal Harmonics
    See also (34.3.22), and for further related integrals see Askey et al. (1986). …
    14.30.8_5 e t a x = 4 π n = 0 m = - n n t n r n λ m Y n , m ( θ , ϕ ) ( 2 n + 1 ) ( n + m ) ! ( n - m ) ! ,
    §14.30(iii) Sums
    14.30.9 P l ( cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos ( ϕ 1 - ϕ 2 ) ) = 4 π 2 l + 1 m = - l l Y l , m ( θ 1 , ϕ 1 ) ¯ Y l , m ( θ 2 , ϕ 2 ) .
    See Andrews et al. (1999, Chapter 9). …
    8: Bibliography M
  • D. R. Masson (1991) Associated Wilson polynomials. Constr. Approx. 7 (4), pp. 521–534.
  • T. Masuda (2003) On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.
  • S. C. Milne (1985c) A new symmetry related to SU ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • D. S. Mitrinović (1970) Analytic Inequalities. Springer-Verlag, New York.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • 9: Bibliography C
  • CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
  • B. C. Carlson (1985) The hypergeometric function and the R -function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.
  • L. Chihara (1987) On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18 (1), pp. 191–207.
  • D. V. Chudnovsky and G. V. Chudnovsky (1988) Approximations and Complex Multiplication According to Ramanujan. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), G. E. Andrews, R. A. Askey, B. C. Bernd, K. G. Ramanathan, and R. A. Rankin (Eds.), pp. 375–472.
  • G. M. Cicuta and E. Montaldi (1975) Remarks on the full asymptotic expansion of Feynman parametrized integrals. Lett. Nuovo Cimento (2) 13 (8), pp. 310–312.
  • 10: 18.17 Integrals
    The case x = 1 is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972). … provided that + m + n is even and the sum of any two of , m , n is not less than the third; otherwise the integral is zero. …