TOPICS
Multiple series generalizations of basic hypergeometric series over the unitary groups 
.
The fundamental theorem of 
series takes 
, ..., 
and 
, ..., 
as indeterminates and 
. Then
![((c_1...c_n;q)_N)/((q;q)_N)
=sum_(y_1,y_2,...,y_n>=0; |y|=N){product_(1<=r<s<=n)[(1-(x_r)/(x_s)q^(y_r-y_s))/(1-(x_r)/(x_s))]×product_(r,s=1)^n[(((x_r)/(x_s)c_s;q)_(y_r))/((q(x_r)/(x_s);q)_(y_r))][q^(y_2+2y_3+...+(n-1)y_n)]},](/images/equations/UnBasicHypergeometricSeries/NumberedEquation1.svg) |
where it is assumed that none of the denominators vanish (Bhatnagar 1995, p. 22). The series in this theorem is called an 
series (Milne 1985; Bhatnagar 1995, p. 22).
Many other 
-results,
including the q-binomial theorem and
q-Saalschütz sum, can be generalized
to 
series.
Basic Hypergeometric Series." Ch. 2 in Inverse
Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis.
Ohio State University, pp. 20-38, 1995.Biedenharn, L. C. and
Louck, J. D. Angular
Momentum in Quantum Physics: Theory and Applications. Reading, MA: Addison-Wesley,
1981.Biedenharn, L. C. and Louck, J. D. The
Racah-Wigner Algebra in Quantum Theory. Reading, MA: Addison-Wesley, 1981.Denis,
R. Y. and Gustafson, R. A. "An 
-Beta Integral Transformation and Multiple Hypergeometric Series
Identities." SIAM J. Math. Anal. 23, 552-561, 1992.Gustafson,
R. A. "Multilateral Summation Theorems for Ordinary and Basic Hypergeometric
Series in 
."
SIAM J. Math. Anal. 18, 1576-1596, 1987.Gustafson, R. A.
and Krattenthaler, C. "Heine Transformations for a New Kind of Basic Hypergeometric
Series in 
."
J. Comput. Appl. Math. 68, 151-158, 1996.Gustafson, R. A.
and Krattenthaler, C. "Determinants Evaluations and 
Extensions of Heine's 
Transformations." In Special
Functions, q-Series, and Related Topics (Ed. M. E. H. Ismail,
D. R. Masson, and M. Rahman). Providence, RI: Amer. Math. Soc., pp. 83-89,
1997.Holman, W. J. III. "Summation Theorems for Hypergeometric
Series in 
."
SIAM J. Math. Anal. 11, 523-532, 1980.Holman, W. J.
III.; Biedenharn, L. C.; and Louck, J. D. "On Hypergeometric Series
Well-Poised in 
." SIAM J. Math. Anal. 7, 529-541, 1976.Milne,
S. C. "An Elementary Proof of the Macdonald Identities for 
." Adv. Math. 57, 34-70, 1985.Milne,
S. C. "Basic Hypergeometric Series Very Well-Poised in 
." J. Math. Anal. Appl. 122, 223-256,
1987.Milne, S. C. "Balanced 
Summation for 
Basic Hypergeometric Series." Adv. Math. 131,
93-187, 1997.Weisstein, Eric W. "U(n) Basic Hypergeometric Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UnBasicHypergeometricSeries.html