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1: 17.12 Bailey Pairs
Bailey Transform
2: 17.10 Transformations of ψ r r Functions
§17.10 Transformations of ψ r r Functions
Bailey’s ψ 2 2 Transformations
3: Bibliography M
  • S. C. Milne and G. M. Lilly (1992) The A l and C l Bailey transform and lemma. Bull. Amer. Math. Soc. (N.S.) 26 (2), pp. 258–263.
  • 4: 17.9 Further Transformations of ϕ r r + 1 Functions
    Bailey’s Transformation of Very-Well-Poised ϕ 7 8
    5: Bibliography
  • G. E. Andrews (2001) Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.
  • 6: Bibliography B
  • W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
  • 7: 16.4 Argument Unity
    Transformations for both balanced F 3 4 ( 1 ) and very well-poised F 6 7 ( 1 ) are included in Bailey (1964, pp. 56–63). …See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …
    8: 16.6 Transformations of Variable
    §16.6 Transformations of Variable
    Quadratic
    Cubic
    16.6.2 F 2 3 ( a , 2 b - a - 1 , 2 - 2 b + a b , a - b + 3 2 ; z 4 ) = ( 1 - z ) - a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a - b + 3 2 ; - 27 z 4 ( 1 - z ) 3 ) .
    For Kummer-type transformations of F 2 2 functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
    9: Bibliography W
  • S. O. Warnaar (1998) A note on the trinomial analogue of Bailey’s lemma. J. Combin. Theory Ser. A 81 (1), pp. 114–118.
  • G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.
  • F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
  • D. V. Widder (1979) The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.
  • D. V. Widder (1941) The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.
  • 10: 17.6 ϕ 1 2 Function
    Bailey–Daum q -Kummer Sum
    Heine’s First Transformation
    Heine’s Third Transformation
    Fine’s Second Transformation
    Fine’s Third Transformation