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Bailey's Transformation

Bailey's transformation is the very general hypergeometric transformation

_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],
(1)

where k=1+2a-b-c-d, and the parameters are subject to the restriction

b+c+d+e+f+g-m=2+3a
(2)

(Bailey 1935, p. 27).

Bhatnagar (1995, pp. 17-18) defines a Bailey transform as follows. Let (a;q)_n be the q-Pochhammer symbol, let a be an indeterminate, and let the lower triangular matrices F=(f)_(nk) and G=(g)_(nk) be defined as

f_(nk)=1/((q;q)_(n-k)(aq;q)_(n+k))
(3)

and

g_(nk)=((1-aq^(2n))(a;q)_(n+k))/((1-a)(q;q)_(n-k))(-1)^(n-k)q^((n-k; 2)).
(4)

Then F and G are matrix inverses.

See also

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. https://web.archive.org/web/20040218044148/http://www.math.ohio-state.edu/~milne/papers/Gaurav.whole.thesis.7.4.ps.Milne, S. C. and Lilly, G. M. "The A_l and C_l 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 Resource. https://mathworld.wolfram.com/BaileysTransformation.html

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