A q-analog of the Gauss summation theorem for hypergeometric series in U(n). (English) Zbl 0658.33005
A q-analog of Holman’s U(n) generalization of the Gauss summation theorem for hypergeometric series is derived. This result is obtained by iterating the author’s generalization of the Gauss reduction formula for basic hypergeometric series in U(n). Multiple series generalizations of both q-analogs of the Chu-Vandermonde summation theorem occur as special cases. Letting \(q\to 1\) the corresponding results for ordinary hypergeometric series in U(n) are obtained. U(n) generalizations the beta integral and Euler’s integral representation of a \({}_ 2F_ 1\) hypergeometric series are also given. Finally, some q-analogs of partial fractions expansions are discussed.
Reviewer: R.A.Gustafson
MSC:
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33B15 | Gamma, beta and polygamma functions |
Digital Library of Mathematical Functions:
§17.15 Generalizations ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related FunctionsReferences:
| [1] | Andrews, G. E., Applications of basic hypergeometric functions, SIAM Rev., 16, 441-484 (1974) · Zbl 0299.33004 |
| [2] | Andrews, G. E., Problems and prospects for basic hypergeometric functions, (Askey, R., Theory and Application of Special Functions (1975), Academic Press: Academic Press New York), 191-224 · Zbl 0342.33001 |
| [3] | Andrews, G. E., (Rota, G.-C, Encyclopedia of Mathematics and Its Applications. Encyclopedia of Mathematics and Its Applications, The Theory of Partitions, Vol. 2 (1976), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0371.10001 |
| [4] | Andrews, G. E., \(q\)-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, (NSF-CBMS Regional Conference Series (1986)), Number 66 · Zbl 0594.33001 |
| [5] | Andrews, G. E.; Askey, R., Enumeration of partitions: The role of Eulerian series and \(q\)-orthogonal polynomials, (Aigner, M., Higher Combinatorics (1977), Reidel: Reidel Dordrecht) · Zbl 0381.10008 |
| [6] | Appéll, P.; Kampéde Fériet, J., Fonctions hypergéometriques et hypersphériques: Polynomes d’Hermites (1926), Gauthier-Villars: Gauthier-Villars Paris · JFM 52.0361.13 |
| [7] | Askey, R., The \(q\)-gamma and \(q\)-beta functions, Appl. Anal., 8, 125-141 (1978) · Zbl 0398.33001 |
| [8] | Askey, R., Book review, Zentralblatt für Mathematik (Mathematics Abstracts), Vol. 514, 161-163 (1984), Review 33001 |
| [9] | R. Askey; R. Askey |
| [10] | Bailey, W. N., Generalized Hypergeometric Series, (Cambridge Mathematical Tract No. 32 (1935), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0011.02303 |
| [11] | Dyson, F. J., Missed opportunities, Bull. Amer. Math. Soc., 78, 635-653 (1972) · Zbl 0271.01005 |
| [12] | H. Flanders; H. Flanders |
| [13] | Greub, W., Linear Algebra (1975), Springer-Verlag: Springer-Verlag New York · Zbl 0317.15002 |
| [14] | Gustafson, R. A., A Whipple’s transformation for hypergeometric series in \(U(n)\) and multivariate hypergeometric orthogonal polynomials, SIAM J. Math. Anal., 18, 495-530 (1987) · Zbl 0607.33015 |
| [15] | Gustafson, R. A.; Milne, S. C., Schur functions, Good’s identity, and hypergeometric series well poised in \(SU (n)\), Adv. in Math., 48, 177-188 (1983) · Zbl 0516.33015 |
| [16] | Gustafson, R. A.; Milne, S. C., A \(q\)-analog of transposition symmetry for invariant \(G\)-functions, J. Math. Anal. Appl., 114, 210-240 (1986) · Zbl 0587.33010 |
| [17] | Hahn, W., Beitrage zur theorie der Heinesehen reihen, die 24 integrale der hypergeometrischen \(q\)-differenzengleichung das \(q\)-analogen der Laplace transformation, Math. Nachr., 2, 263-278 (1949) · Zbl 0033.05702 |
| [18] | Hahn, W., Über die hoheren Heineschen Rechen und eine einheitliche Theorie der Sogennanten speziellen Funktionen, Math. Nachr., 3, 257-294 (1950) · Zbl 0038.05002 |
| [19] | Heine, E., Untersuchungen über die Reihe…, J. Reine Angew. Math., 34, 285-328 (1847) · ERAM 034.0971cj |
| [20] | Holman, W. J., Summation theorems for hypergeometric series in \(U(n)\), SIAM J. Math. Anal., II, 523-532 (1980) · Zbl 0454.33010 |
| [21] | Holman, W. J.; Biedenharn, L. C.; Louck, J. D., On hypergeometric series well-poised in \(SU (n)\), SIAM J. Math. Anal., 7, 529-541 (1976) · Zbl 0329.33013 |
| [22] | Horn, J., Über die Convergenz der hypergeometrische Reihen zweier und dreier Veränderlichen, Math. Ann., 34, 544-600 (1889) · JFM 21.0449.02 |
| [23] | Ismail, M. E.H, A simple proof of Ramanujan’s \(_1Ψ_1\) sum, (Proc. Amer. Math. Soc., 63 (1977)), 185-186 · Zbl 0351.33002 |
| [24] | Jackson, F. H., On basic double hypergeometric functions, Quart. J. Math. Oxford, 13, 69-82 (1942) · Zbl 0060.19809 |
| [25] | Jackson, F. H., Basic double hypergeometric functions (II), Quart. J. Math. Oxford, 15, 49-61 (1944) · Zbl 0060.19810 |
| [26] | Kac, V. G., Infinite Dimensional Lie Algebras, (Progress in Mathematics, Vol. 44 (1983), Birkhäuser: Birkhäuser Boston) · Zbl 0425.17009 |
| [27] | K. Leeb; K. Leeb |
| [28] | Lepowsky, J.; Milne, S., Lie algebraic approaches to classical partition identities, Adv. in Math., 29, 15-59 (1978) · Zbl 0384.10008 |
| [29] | Louck, J. D., Theory of Angular Momentum in \(N\)-Dimensional Space, Los Alamos Scientific Laboratory Report LA-2451 (1960), and the cited contribution of E. D. Cashwell |
| [30] | Louck, J. D.; Biedenharn, L. C., Canonical unit adjoint tensor operators in \(U(n)\), J. Math. Phys., 11, 2368-2414 (1970) · Zbl 0196.14501 |
| [31] | Macdonald, I. G., Affine root systems and Dedekind’s \(n\)-function, Invent. Math., 15, 91-143 (1972) · Zbl 0244.17005 |
| [32] | Macdonald, I. G., Symmetric Functions and Hall Polynomials (1979), Oxford Univ. Press: Oxford Univ. Press London/New York · Zbl 0487.20007 |
| [33] | I. G. Macdonald; I. G. Macdonald |
| [34] | Mathai, A. M.; Saxena, R. K., Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, (Lecture Notes in Mathematics, Vol. 348 (1973), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0272.33001 |
| [35] | Miller, W., Lie theory and \(q\)-difference equations, SIAM J. Math. Anal., 1, 2, 171-178 (1970) · Zbl 0206.39001 |
| [36] | Milne, S. C., Hypergeometric series well-poised in \(SU (n)\) and a generalization of Biedenharn’s \(G\) functions, Adv. in Math., 36, 169-211 (1980) · Zbl 0451.33010 |
| [37] | Milne, S. C., A new symmetry related to \(SU (n)\) for classical basic hypergeometric series, Adv. in Math., 57, 71-90 (1985) · Zbl 0586.33012 |
| [38] | Milne, S. C., A \(q\)-analog of hypergeometric series well-poised in \(SU (n)\) and invariant \(G\)-functions, Adv. in Math., 58, 1-60 (1985) · Zbl 0586.33014 |
| [39] | Milne, S. C., A \(q\)-analog of the \(_5F_4(1)\) summation theorem for hypergeometric series well-poised in \(SU (n)\), Adv. in Math., 57, 14-33 (1985) · Zbl 0586.33010 |
| [40] | Milne, S. C., An elementary proof of the Macdonald identities for \(A^{(1)}_l\), Adv. in Math., 57, 34-70 (1985) · Zbl 0586.33011 |
| [41] | Milne, S. C., A \(U(n)\) generalization of Ramanujan’s \(_1Ψ_1\) summation, J. Math. Anal. Appl., 118, 263-277 (1986) · Zbl 0594.33008 |
| [42] | Milne, S. C., Basic hypergeometric series very well poised in \(U(n)\), J. Math. Anal. Appl., 122, 223-256 (1987) · Zbl 0607.33014 |
| [43] | Rainville, E. D., Special Function (1960), Macmillan Co: Macmillan Co New York · Zbl 0092.06503 |
| [44] | Slater, L. J., Generalized Hypergeometric Functions (1966), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0135.28101 |
| [45] | V. Strehl; V. Strehl |
| [46] | Whipple, F. J.W, On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc., 24, 2, 247-263 (1926) · JFM 51.0283.03 |
| [47] | Whipple, F. J.W, Well-poised series and other generalized hypergeometric series, Proc. London Math. Soc., 25, 2, 525-544 (1926) · JFM 52.0365.03 |
| [48] | Whittaker, E. T.; Watson, G. N., Modern Analysis (1927), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0108.26903 |
| [49] | J. Wimp, private communication, May; J. Wimp, private communication, May |
| [50] | Winquist, L., An elementary proof of \(p(11m + 6)\) ≡ 0(mod 11), J. Combin. Theory, 6, 56-59 (1969) · Zbl 0241.05006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
