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A q-analog of hypergeometric series well-poised in SU(n) and invariant G-functions. (English) Zbl 0586.33014
We introduce natural q-analogs of hypergeometric series well-poised in SU(n), the related hypergeometric series in U(n), and invariant G-functions. We prove that both the SU(n) multiple q-sereis and the invariant G-functions satisfy general q-difference equations. Both the SU(N) and U(n) q-series are new multivariable generalizations of classical basic hypergeometric series of one variable. We prove an identity which expresses our U(n) multiple q-series as a finite sum of finite products of classical basic hypergeometric series. These U(n) q- sereis also satisfy an elegant reduction formula which is analogous to the ”inclusion lemma” for classical invariant G-functions.

##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras 33C60 Hypergeometric integrals and functions defined by them 33C05 Classical hypergeometric functions, ${}_2F_1$ 22E70 Applications of Lie groups to physics; explicit representations
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##### References:
 [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. Theory and application of special functions, 191-224 (1975) [3] Andrews, G. E.: G.-crotaencycopedia of mathematics & its applications. The theory of partitions 2 (1976) [4] Appéll, P.; De Fériet, J. Kampé: Fonctions hypergéométriques et hypersphériques: polynomes d’hermites. (1926) [5] Bailey, W. N.: Generalized hypergeometric series. Cambridge mathematical tract no. 32 (1935) · Zbl 0011.02303 [6] Biedenharn, L.; Iii, W. Holman; Milne, S.: The invariant polynomials characterizing $U(n)$ tensor operators \langlep, q,$\dots , q, 0,\dots , 0\rangle$having maximal null space. Adv. in appl. Math. 1, 390-472 (1980) · Zbl 0457.33010 [7] Biedenharn, L. C.; Gustafson, R. A.; Milne, S. C.: An umbral calculus for polynomials characterizing $U(n)$ tensor operators. Adv. in math. 51, 36-90 (1984) · Zbl 0534.33010 [8] Biedenharn, L. C.; Louck, J. D.: A pattern calculus for tensor operators in the unitary groups. Comm. math. Phys. 8, 89-131 (1968) · Zbl 0155.32405 [9] Biedenharn, L. C.; Louck, J. D.; Chàcon, E.; Ciftan, M.: On the structure of the canonical tensor operators in the unitary groups. I. an extension of the pattern calculus rules and the canonical splitting in $U(3)$. J. math. Phys. 13, 1957-1984 (1972) · Zbl 0253.20069 [10] Biedenharn, L. C.; Louck, J. D.: On the structure of the canonical tensor operators in the unitary groups. II. the tensor operators in $U(3)$ characterized by the maximal nul space. J. math. Phys. 13, 1985-2001 (1972) · Zbl 0253.20070 [11] Biedenharn, L. C.; Louck, J. D.: G.-crotaencyclopedia of mathematics & its applications. Angular momentum in quantum physics: theory and applications 8 (1981) · Zbl 0474.00023 [12] Biedenharn, L. C.; Louck, J. D.: G.-crotaencyclopedia of mathematics & its applications. The racah-Wigner algebra in quantum theory 9 (1981) · Zbl 0474.00023 [13] Biedenharn, L. C.; Gustafson, R. A.; Lohe, M. A.; Louck, J. D.; Milne, S. C.: Special functions and group theory in theoretical physics. Proceedings, mathematisches forschunginstitut oberwolfach, 129-162 (13-19 March 1983) [14] L. C. Biedenharn, R. A. Gustafson, and S. C. Milne, U(n) Wigner coefficients, the path sum formula and invariant G-functions, Adv. in Appl. Math., in press. · Zbl 0586.33015 [15] Exton, H.: Multiple hypergeometric functions and applications. (1976) · Zbl 0337.33001 [16] Exton, H.: Handbook of hypergeometric integrals: theory, applications, tables, computer programs. (1978) · Zbl 0377.33001 [17] Exton, H.: Q-hypergeometric functions and applications. (1983) · Zbl 0514.33001 [18] Good, I. J.: Short proof of a conjecture of Dyson. J. math. Phys. 11, 1884 (1970) · Zbl 0194.05903 [19] Greub, W.: Linear algebra. (1975) · Zbl 0317.15002 [20] Gustafson, R. A.; Milne, S. C.: Schur functions and the invariant polynomials characterizing $U(n)$ tensor operators. Adv. in appl. Math. 4, 422-478 (1983) · Zbl 0534.33009 [21] 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 [22] R. A. Gustafson and S. C. Milne, A new symmetry for Biedenharn’s G-functions and classical hypergeometric series, Adv. in Math., in press. · Zbl 0586.33013 [23] R. A. Gustafson and S. C. Milne, A q-analog of transposition symmetry for invariant G-functions, J. Math. Anal. Appl., in press. · Zbl 0587.33010 [24] Hahn, W.: Beitrage zur theorie der heineschen reihen, die 24 integrale der hypergeometrischen q-differenzengleichung, das q-analogen der Laplace transformation. Math. nachr. 2, 263-278 (1949) · Zbl 0033.05703 [25] Hahn, W.: Über die höheren heineschen reihen und eine einheitliche theorie der sogenannten speziellen funktionen. Math. nachr. 3, 257-294 (1950) · Zbl 0038.05002 [26] Hildebrand, F.: Introduction to numerical analysis. (1956) · Zbl 0070.12401 [27] Iii, W. J. Holman: Summation theorems for hypergeometric series in $U(n)$. SIAM J. Math. anal. 11, 523-532 (1980) · Zbl 0454.33010 [28] Iii, W. J. Holman; 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 [29] Horn, J.: Ueber die convergenz der hypergeometrische reihen zweier und dreier veränderlichen. Math. ann. 34, 544-600 (1889) [30] Louck, J. D.: Theory of angular momentum in N-dimensional space. Los alamos scientific laboratory report LA-2451 (1960) [31] Louck, J. D.; Biedenharn, L. C.: On the structure of the canonical tensor operators in the unitary groups. III. further developments of the boson polynomials and their implications. J. math. Phys. 14, 1336-1357 (1973) [32] Macdonald, I. G.: Symmetric functions and Hall polynomials. (1979) · Zbl 0487.20007 [33] Mathai, A. M.; Saxena, R. K.: Generalized hypergeometric functions with applications in statistics and physical sciences. Lecture notes in mathematics 348 (1973) · Zbl 0272.33001 [34] 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 [35] S. C. Milne, A new symmetry related to SU(n) for classical basic hypergeometric series, Adv. in Math., in press. · Zbl 0586.33012 [36] S. C. Milne, A q-analog of the 5F4(1) summation theorem for hypergeometric series well-poised in SU(n), Adv. in Math., in press. · Zbl 0586.33010 [37] S. C. Milne, An elementary proof of the Macdonald identities for A(1)l, Adv. in Math., in press. · Zbl 0586.33011 [38] Rainville, E. D.: Special functions. (1960) · Zbl 0092.06503 [39] Slater, L. J.: Generalized hypergeometric functions. (1966) · Zbl 0135.28101 [40] Wilson, K.: Proof of a conjecture by Dyson. J. math. Phys. 3, 1040-1043 (1962) · Zbl 0113.21403