TOPICS
Search

q-Gauss Identity

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

A q-analog of Gauss's theorem due to Jacobi and Heine,

_2phi_1(a,b;c;q,c/(ab))=((c/a;q)_infty(c/b;q)_infty)/((c;q)_infty(c/(ab);q)_infty)
(1)

for |c/(ab)|<1 (Gordon and McIntosh 1997; Koepf 1998, p. 40), where _2phi_1(a,b;c;q,z) is a q-hypergeometric function. A special case for a=q^(-n) is given by

sum_(k=0)^(n)q^(k^2)[n; k]_q^2=((sqrt(q);q)_n(-sqrt(q);q)_n(-q;q)_n)/((q;q)_n)
(2)
=((-q;q)_n(q;q^2)_n)/((q;q)_n),
(3)

where [n; k]_q is a q-binomial coefficient (Koepf 1998, p. 43).

See also

q-Chu-Vandermonde Identity, q-Hypergeometric Function

Explore with Wolfram|Alpha

References

Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 31, 1995.Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 10 and 236, 1990.Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431-448, 1997.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.

Referenced on Wolfram|Alpha

q-Gauss Identity

Cite this as:

Weisstein, Eric W. "q-Gauss Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-GaussIdentity.html

Subject classifications