Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Inversion of bilateral basic hypergeometric series | Inversion of bilateral basic hypergeometric series Michael Schlosser Institut fur Mathematik der Universitat Wien Strudlhofgasse 4 A-1090 Wien Austria schlosse@ Submitted Jun 5 2002 Accepted Jan 8 2003 Published Mar 18 2003 MR Subject Classifications 33D15 15A09 Abstract We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey s very-well-poised 6W6 summation theorem and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive via inverse relations several new identities for bilateral basic hypergeometric series. 1 Introduction Bailey s 10 Eq. very-well-poised 6Ộ6 summation formula 6 6 qựa -qựa b c d e a2q a -ựa aq b aq c aq d aq e q bcde q aq q a aq bc aq bd aq be aq cd aq ce aq de q aq b aq c aq d aq e q b q c q d q e a2q bcde q where a2q bcde 1 cf. 19 Eq. stands on the top of the classical hierarchy of summation theorems for bilateral basic hypergeometric series. It contains many important identities as special cases among them Jacobi s triple product identity the q-Pfaff-Saalschutz summation and the q-binomial theorem to name just a few. Various applications of Bailey s 6 6 summation exist in number theory see Andrews 4 pp. 461-468 and in special functions see . Ismail and Masson 23 . A combinatorial partition theoretic application of Bailey s 6 6 summation formula was recently revealed in remarkable work of Alladi Andrews and Berkovich 2 . Supported by an APART grant of the Austrian Academy of Sciences THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R10 1 Different proofs of are known. A very elegant proof using analytic continuation was given by Askey and Ismail 9 . For an elementary proof using manipulations of series see Schlosser 38 . In addition to Bailey s 6 6 summation formula there is a .
