On some properties of generalized Narayana numbers

Abstract

In this paper, we consider a three-parameter generalization of the Narayana numbers that is related to the powers of the generating function for the  Narayana numbers. We find a multivariate generating function for these generalized Narayana numbers. For the composition of generating functions where the inner function is the generating function for the Narayana numbers, we obtain an explicit formula for calculating its coefficients. We also consider several examples of applying the obtained results and derive new generating functions.