Q-Analogues Of Some Combinatorial Congruences

Congruences on sums involving combinatorial numbers have been received much attention in recent years.N.J.Calkin,L.Van Hamme,F.Rodriguez-Villegas,W.Zudilin,L.Long,Z.-W.Sun,Z.-H.Sun,J.Zeng and V.J.W.Guo have provided a series of congruences related to sums of binomial coefficients and made many interesting conjectures.Besides,some results on q-congruences have been given by R.Tauraso,A.Straub,H.Pan and V.J.W.Guo,respective-ly.It is a relatively new research subject to study q-analogues of combinatorial congruences.By using q-series identities and induction,we give q-analogues of some divisi-bility results on factors of sums involving binomial coefficients obtained by V.J.W.Guo and J.Zeng.We also give some congruences on sums involving q-Catalan numbers,q-super Catalan numbers,q-Narayana numbers and q-ballot numbers.Applying Zeilberger's algorithm,we partially settle a recent congruence conjec-ture of V.J.W.Guo on sums involving(k2k)4.Moreover,we give a q-analogue a superconruence modulo p4 of L.Long.As a conclusion,we prove the following result observed by L.Long:(?)4k+1/256k(k2k)4?pr(mod pr+3)where p? 5 is a prime and r is a positive integer.