Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices

This work is based on the Euler-Seidel matrix method [11] which is related to algorithms, combinatorics and generating functions. This method is quite useful to investigate properties of some special numbers and polynomials. In this paper we consider the Euler-Seidel matrix method for some combinatorial numbers and polynomials. This method is relatively easier than the most of combinatorial methods to investigate the structure of such numbers and polynomials. We obtain new properties of geometric (or Fubini) polynomials and numbers. In addition, we use this method to find out some equations and recurrence relations of exponential (or Bell) polynomials and numbers, and Stirling numbers of the second kind. Although some results in this paper are known, this method provides different proofs as well as new identities. We first consider a given sequence (an)n≥0, and then determine the Euler-Seidel matrix corresponding to this sequence is recursively by the formulae