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
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.
![_9F_8[a, 1+1/2a, b, c, d,; 1/2a, 1+a-b, 1+a-c, 1+a-d,
e, f, g, -m;; 1+a-e, 1+a-f, 1+a-g, 1+a+m]
=((1+a)_m(1+k-e)_m(1+k-f)_m(1+k-g)_m)/((1+k)_m(1+a-e)_m(1+a-f)_m(1+a-g)_m)_9F_8[k, 1+1/2k, k+b-a, k+c-a, k+d-a,; 1/2k, 1+a-b, a+a-c, 1+a-d,e, f, g, -m;; 1+k-e, 1+k-f, 1+k-g, 1+k+m],](/images/equations/BaileysTransformation/NumberedEquation1.svg)
,
and the parameters are subject to the restriction
be the q-Pochhammer
symbol, let 
be an indeterminate, and let the lower triangular
matrices 
and 
be defined as
and 
are matrix inverses.
and 
Bailey Transform and Lemma." Bull. Amer. Math. Soc. 26,
258-263, 1992.