\(q\)-Triplicate inverse series relations with applications to \(q\)-series. (English) Zbl 1087.33008
Inverse relations were investigated in depth in [J. Riordan, An introduction to combinatorial analysis. Wiley (1958; Zbl 0078.00805)]. A special case is the identities of H. W. Gould and L. C. Hsu [“Some new inverse series relations”, Duke Math J. 40, 885–891 (1973; Zbl 0281.05008)], and its \(q\)-analogue [L. Carlitz, “Some inverse relations”, Duke Math J. 40, 893–901 (1973; Zbl 0276.05012)].
The paper under review treats a \(q\)-triplicate inverse relation in the spirit of Carlitz [loc. cit.], which is a \(q\)-analogue of papers by Chu. Various special cases of this inverse relation are studied, however the reviewer is uncertain about the validity of formulas 2.10-12, 2.14-16 and later formulas. Anyway the matrix inverse relation is very interesting.
The paper under review treats a \(q\)-triplicate inverse relation in the spirit of Carlitz [loc. cit.], which is a \(q\)-analogue of papers by Chu. Various special cases of this inverse relation are studied, however the reviewer is uncertain about the validity of formulas 2.10-12, 2.14-16 and later formulas. Anyway the matrix inverse relation is very interesting.
Reviewer: Thomas Ernst (Uppsala)
MSC:
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
| 15A09 | Theory of matrix inversion and generalized inverses |
References:
| [1] | L. Carlitz, Some inverse relations , Duke Math. J. 40 (1973), 45-50. · Zbl 0276.05012 · doi:10.1215/S0012-7094-73-04083-0 |
| [2] | W. Chu, Inversion techniques and combinatorial identities : Balanced hypergeometric series , Rocky Mountain J. Math. 32 (2002), 561-587. · Zbl 1038.33002 · doi:10.1216/rmjm/1030539687 |
| [3] | ——–, Inversion techniques and combinatorial identities , Boll. Un. Mat. Ital. 7 (1993), 737-760. · Zbl 0816.33003 |
| [4] | ——–, Inversion techniques and combinatorial identities : A unified treatment for the \(_7F_6\)-series identities , Collect. Math. 45 (1994), 13-43. · Zbl 0844.33006 |
| [5] | ——–, Basic hypergeometric identities : An introductory revisiting through the Carlitz inversions , Forum Math. 7 (1995), 117-129. · Zbl 0815.05009 · doi:10.1515/form.1995.7.117 |
| [6] | G. Gasper and M. Rahman, Basic hypergeometric series , Encyclopedia Math. Appl., vol. 35, Cambridge Univ. Press, Cambridge, 1990. · Zbl 0695.33001 |
| [7] | I.M. Gessel and D. Stanton, Application of \(q\)-Lagrange inversion to basic hypergeometric series , Trans. Amer. Math. Soc. 277 (1983), 173-203. JSTOR: · Zbl 0513.33001 · doi:10.2307/1999351 |
| [8] | ——–, Another family of \(q\)-Lagrange inversion formulas , Rocky Mountain J. Math. 16 (1986), 373-384. · Zbl 0613.33003 · doi:10.1216/RMJ-1986-16-2-373 |
| [9] | H.W. Gould and L.C. Hsu, Some new inverse series relations , Duke Math. J. 40 (1973), 885-891. · Zbl 0281.05008 · doi:10.1215/S0012-7094-73-04082-9 |
| [10] | L.J. Slater, A new proof of Rogers’s transformations of infinite series , Proc. London Math. Soc. 2 (1951), 460-475. · Zbl 0044.06102 · doi:10.1112/plms/s2-53.6.460 |
| [11] | A. Verma and V. K. Jain, Transformations between basic hypergeometric series on different bases and identities of Rogers-Ramanujan type , J. Math. Anal. Appl. 76 (1980), 230-269. · Zbl 0443.33004 · doi:10.1016/0022-247X(80)90076-1 |
| [12] | Yusen Zhang and Tianming Wang, Applications of Chu’s \(q\)-duplicate inverse series relations , submitted. · Zbl 1165.33012 |
| [13] | Yusen Zhang and Mingshu Tan, Summation formulae including a \(_3\phi_2\) series , J. Southwest China Norm. Univ. (Natural Science) 5 (2002), 646-651. |
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