The number of convex polyominoes and the generating function of Jacobi polynomials

Lin and Chang gave a generating function of convex polyominoes with an m+1 by n + 1 minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is m+ n+mn m+ n ( 2m+ 2n 2m ) − 2mn m+ n ( m+ n m )2 . We show that this result can be derived from some binomial coefficients identities related to the generating function of Jacobi polynomials. Some (binomial coefficients) identities arise from alternative solutions of combinatorial problems and incidentally give added significance to doing problems the “hard” way.