Some \(q\)-analogs of congruences for central binomial sums (Q2866752)

Four congruences, with a \(q\)-counterpart and involving central binomial coefficients, are investigated through the method introduced by \textit{D. Zeilberger} [Contemp. Math. 143, 579--607 (1993; Zbl 0808.05010)] and refined by \textit{M. Mohammed} [Ann. Comb. 9, No. 2, 205--221 (2005; Zbl 1078.33022)].NEWLINENEWLINEBeyond the \(q\)-WZ method, the proof employs: a theorem given by \textit{L.-L. Shi} and \textit{H. Pan} [Am. Math. Mon. 114, No. 6, 529--531 (2007; Zbl 1193.11018)], a congruence provided by \textit{Z.-W. Sun} and \textit{R. Tauraso} [Adv. Appl. Math. 45, No. 1, 125--148 (2010; Zbl 1231.11021)], an identity supplied by \textit{G. E. Andrews} [SIAM Rev. 16, 441--484 (1974; Zbl 0299.33004)] and a sum explored by both \textit{K. Dilcher} [Electron. J. Comb. 15, No. 1, Research Paper R63, 18 p. (2008; Zbl 1206.11024)] and \textit{H. Pan} [Acta Arith. 128, No. 4, 303--318 (2007; Zbl 1138.11005)].NEWLINENEWLINEThe author remarks that the new \(q\)-congruences are not related to those already found by himself ([Adv. Appl. Math. 48, No. 5, 603--614 (2012; Zbl 1270.11016)]) or by others ([\textit{V. J. W. Guo} and \textit{J. Zeng}, Adv. Appl. Math. 45, No. 3, 303--316 (2010; Zbl 1231.11020)]) except for the connections with a result of \textit{G. E. Andrews} [Discrete Math. 204, No. 1--3, 15--25 (1999; Zbl 0937.05014)] and with the irrationality of the \(q\)-series studied by \textit{T. Amdeberhan} and \textit{D. Zeilberger} [Adv. Appl. Math. 20, No. 2, 275--283 (1998; Zbl 0914.11042)].NEWLINENEWLINEThe author also suggests the search for further \(q\)-analogs like, \(e.g.\), for a congruence derived by the same \textit{R. Tauraso} [J. Number Theory 130, No. 12, 2639--2649 (2010; Zbl 1208.11027)] from a work of \textit{Kh. Hessami Pilehrood} and \textit{T. Hessami Pilehrood} [J. Symb. Comput. 46, No. 6, 699--711 (2011; Zbl 1254.11023)].