Symmetric recurrence relations and binomial transforms

We consider a family of sums which satisfy symmetric recurrence relations. A sufficient and necessary condition for the existence of such recurrence relations is given. Let us call a pair of sequence (an,bn)(an,bn) a binomial pair if anan is the binomial transform of bnbn. We give some ways of constructing new binomial pairs from old ones. We further generalize the binomial transform by adding a parameter and show that the generalized binomial transform is an involution.