In this work we study the period function T of solutions to the conservative equationWe present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H (x, y) = F (x) + n −1 |y| n . Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation u = u −1 in two dimensions.

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We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G.E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a given Riordan array, by the elimination of elements. We extend the method and as an application we obtain other identities, some of which are new. An important feature of our construction is that it establishes a nice connection between the generating function of the A-sequence of a certain class of Riordan arrays and hypergeometric functions.