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Watson-Whipple Transformation

If at least one of d, e, or f has the form q^(-N) for some nonnegative integer N (in which case both sums terminate after N+1 terms), then

_8phi_7[a,qa^(1/2),-qa^(1/2),b,c,d,e,f; a^(1/2),-a^(1/2),(aq)/b,(aq)/c,(aq)/d,(aq)/e,(aq)/f;q,(a^2q^2)/(bcdef)] =((aq,(aq)/(de),(aq)/(df),(aq)/(ef))_infty)/(((aq)/d,(aq)/e,(aq)/f,(aq)/(def))_infty)_4phi_3[(aq)/(bc),d,e,f; (aq)/b,(aq)/c,(def)/a;q,q],

where (a_1,a_2,...,a_r;q)_infty is a generalized q-Pochhammer symbol

(a_1,a_2,...,a_r;q)_infty=(a_1;q)_infty(a_2;q)_infty...(a_r;q)_infty,

and each of _8phi_7 and _4phi_3 is a q-hypergeometric function.

See also

q-Hypergeometric Function, q-Pochhammer Symbol, q-Series

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References

Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 242, 1990.Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." J. London Math. Soc. 62, 321-335, 2000.

Referenced on Wolfram|Alpha

Watson-Whipple Transformation

Cite this as:

Weisstein, Eric W. "Watson-Whipple Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Watson-WhippleTransformation.html

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