Iterative processes related to Riordan arrays: The reciprocation and the inversion of power series

Some parts of the material in this paper were presented by the author in the meeting Combinatorics 2008 celebrated at Costermano, Italy the last June. So it contains some expository on previous work but it also contains new and original results, mainly those in the last two sections. The results in this paper are consequences of a special interpretation as fixed point problems of the two classical reversion processes in the realm of formal power series: the reciprocation, i. e. the reversion for the Cauchy product, and the inversion, i. e. the reversion for the composition of series. The aim of this paper is to show how the (Picard) successive approximation method induced by Banach’s Fixed Point Theorem (and some mild generalization) allows us to treat some mathematical methods in combinatorics. In particular: 1) To obtain a special Riordan array as remainder in a natural approximation problem about the reciprocation of a quadratic polynomial related to the sum of the arithmeticgeometric series. 2) To construct all elements in the Riordan group as a consequence of the iterative process obtained to calculate the reciprocal of any power series admitting it.