A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…}l,m,n∈{0,1,2,…} then∑k=0l(-1)m-klkm-kn2kk-2l+m=∑k=0llk2knn-lm+n-3k-l.On the basis of this identity, for d,r∈{0,1,2,…}d,r∈{0,1,2,…} we construct explicit F(d,r)F(d,r) and G(d,r)G(d,r) such that for any prime p>max{d,r}p>max{d,r} we have∑k=1p-1krCk+d≡F(d,r)(modp)ifp≡1(mod3),G(d,r)(modp)ifp≡2(mod3),where CnCn denotes the Catalan number (2nn)/(n+1. For example, when p⩾5p⩾5 is a prime, we have∑k=1p-1k2Ck≡-23(modp)ifp≡1(mod3),-13(modp)ifp≡2(mod3),and∑0