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U(n) Basic Hypergeometric Series

المؤلف:  Biedenharn, L. C. and Louck, J. D.

المصدر:  Angular Momentum in Quantum Physics: Theory and Applications. Reading, MA: Addison-Wesley, 1981.

الجزء والصفحة:  ...

10-6-2019

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U(n) Basic Hypergeometric Series

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

{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)]}, " src="http://mathworld.wolfram.com/images/equations/UnBasicHypergeometricSeries/NumberedEquation1.gif" style="height:122px; width:453px" />

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.

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REFERENCES:

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.