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Bailey transformation of very-well-poised 8ϕ7

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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
2: 16.4 Argument Unity
Rogers–Dougall Very Well-Poised Sum
Dougall’s Very Well-Poised Sum
Transformations for both balanced F 3 4 ( 1 ) and very well-poised F 6 7 ( 1 ) are included in Bailey (1964, pp. 56–63). A similar theory is available for very well-poised F 8 9 ( 1 ) ’s which are 2-balanced. …
3: 17.9 Further Transformations of ϕ r r + 1 Functions
§17.9 Further Transformations of ϕ r r + 1 Functions
F. H. Jackson’s Transformations
Bailey’s Transformation of Very-Well-Poised ϕ 7 8
Sears–Carlitz Transformation
Mixed-Base Heine-Type Transformations
4: 17.4 Basic Hypergeometric Functions
It is slightly at variance with the notation in Bailey (1964) and Slater (1966). … The series (17.4.1) is said to be very-well-poised when r = s , (17.4.11) is satisfied, and …
5: 17.12 Bailey Pairs
§17.12 Bailey Pairs
Bailey Transform
Bailey Pairs
Weak Bailey Lemma
Strong Bailey Lemma
6: 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).
7: 17.10 Transformations of ψ r r Functions
§17.10 Transformations of ψ r r Functions
Bailey’s ψ 2 2 Transformations
Other Transformations
17.10.3 ψ 8 8 ( q a 1 2 , q a 1 2 , c , d , e , f , a q n , q n a 1 2 , a 1 2 , a q / c , a q / d , a q / e , a q / f , q n + 1 , a q n + 1 ; q , a 2 q 2 n + 2 c d e f ) = ( a q , q / a , a q / ( c d ) , a q / ( e f ) ; q ) n ( q / c , q / d , a q / e , a q / f ; q ) n ψ 4 4 ( e , f , a q n + 1 / ( c d ) , q n a q / c , a q / d , q n + 1 , e f / ( a q n ) ; q , q ) ,
17.10.5 ( a q / b , a q / c , a q / d , a q / e , q / ( a b ) , q / ( a c ) , q / ( a d ) , q / ( a e ) ; q ) ( f a , g a , f / a , g / a , q a 2 , q / a 2 ; q ) ψ 8 8 ( q a , q a , b a , c a , d a , e a , f a , g a a , a , a q / b , a q / c , a q / d , a q / e , a q / f , a q / g ; q , q 2 b c d e f g ) = ( q , q / ( b f ) , q / ( c f ) , q / ( d f ) , q / ( e f ) , q f / b , q f / c , q f / d , q f / e ; q ) ( f a , q / ( f a ) , a q / f , f / a , g / f , f g , q f 2 ; q ) ϕ 7 8 ( f 2 , q f , q f , f b , f c , f d , f e , f g f , f , f q / b , f q / c , f q / d , f q / e , f q / g ; q , q 2 b c d e f g ) + idem ( f ; g ) .
8: 17 q-Hypergeometric and Related Functions
9: Bibliography B
  • D. H. Bailey (1995) A Fortran-90 based multiprecision system. ACM Trans. Math. Software 21 (4), pp. 379–387.
  • W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
  • W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
  • W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.
  • W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.
  • 10: Bibliography W
  • X. Wang and A. K. Rathie (2013) Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.
  • 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.