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# 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.

Dougall-Ramanujan Identity, Generalized Hypergeometric Function, Gould and Hsu Matrix Inversion Formula

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## 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.

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Bailey's Transformation

## Cite this as:

Weisstein, Eric W. "Bailey's Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BaileysTransformation.html