On contiguity relations of Jackson's basic hypergeometric series \(\Upsilon_ 1(a;b;c;x,y,1/2)\) and its generalizations (Q1355887)

This paper proposes a \(q\)-deformation of the Lauricella hypergeometric series of confluent type, which includes a \(q\)-analog of Kummer's hypergeometric series and Jackson's basic double hypergeometric series, and a \(q\)-analog of confluent hypergeometric system derived from these series. The authors defined \(q\)-difference operators, called raising operators and lowering operators that describe the symmetry of the system. The main result is that the contiguity relations of the system is a representation of a \(q\)-deformation of enveloping algebra of semi-direct product of \(sl_{n-2}\) and a finite Heisenberg algebra with \(2n-2\) parameters. Refer to the papers [\textit{I. M. Gel'fand, V. S. Retakh}, and \textit{V. V. Serganova}, Sov. Math. Dokl. 37, No. 1, 8-12 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 1, 17-21 (1988; Zbl 0699.33012)] and [\textit{H. Kimura, Y. Haraoka}, and \textit{K. Takano}, Proc. Japan Acad., Ser. A68, No. 9, 290-295 (1992; Zbl 0773.33004)] for the generalized confluent hypergeometric systems in ordinary sense and to the paper [\textit{M. Noumi}, CWI Q. 5, No. 4, 293-307 (1992; Zbl 0782.33015)] for \(q\)-analogues of Lauricella's hypergeometric series.