On interpolation of some recurrent sequences

We consider the problem of interpolation of recurrent functions given at integer points. Interpolation is understood as a construction of an analytic function that takes given values at given points. In a special case of iterations of analytic functions, i.e., of composition of analytic maps, this is a classical problem of construction of continuous iterations (compositions) of maps, and it is considered as solved. However, the existing methods of construction of such maps are extremely cumbersome technically as well as very complicated with regard to the means used for their proof. We give two elementary methods of solution of this problem which are far superior in efficiency than the existing ones. In particular, we obtain a simple algorithm for inversion of a formal power series (Lagrange formula), which is applicable for more general power-logarithmic series. Additionally, we consider a problem of asymptotics of a recurrent sequence.