Multiple series generalizations of basic hypergeometric series over the unitary groups .
The fundamental theorem of series takes , ..., and , ..., as indeterminates and . Then

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).

Bhatnagar, G. " 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.