Some Properties On The Q-delannoy Numbers

In the first part,we give an algebraic proof of the Lucas-type congruence for q-Delannoy numbers,and give an algebraic proof of a recursion for the q-Delannoy numbers.In the second part,we give a summation formula for a product of two q-Delannoy numbers and use it to prove some congruences for sums involving q-Delannoy numbers.This confirms three recent conjectures of V.J.W.Guo.We further generalize this summation formula by showing that certain terminating 6?5 series can be factorized into a product of two 3?2 series.This generalization may be deemed a q-analogue of the terminating case of a summation formula in Slater's book.