Bailey's Transformation
Bailey's transformation is the very general hypergeometric transformation
 |
(1)
|
where 
, and the parameters are subject to the restriction
 |
(2)
|
(Bailey 1935, p. 27).
Bhatnagar (1995, pp. 17-18) defines a Bailey transform as follows. Let 
be the q-Pochhammer symbol, let 
be an indeterminate, and let the lower triangular matrices 
and 
be defined as
 |
(3)
|
and
 |
(4)
|
Then 
and 
are matrix inverses.
REFERENCES:
Bailey, W. N. "Some Identities Involving Generalized Hypergeometric Series." Proc. London Math. Soc. 29, 503-516, 1929.
Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: University Press, 1935.
Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, 1995. http://www.math.ohio-state.edu/~milne/papers/Gaurav.whole.thesis.7.4.ps.
Milne, S. C. and Lilly, G. M. "The 
and 
Bailey Transform and Lemma." Bull. Amer. Math. Soc. 26, 258-263, 1992.
