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.