Generalized Bernoulli Numbers and Polynomials in the Context of the Clifford Analysis

The Bernoulli polynomials for natural values of the argument were first considered by J. Bernoulli (1713) in relation to the problem of summation of powers of consecutive natural
numbers. L. Euler studied such polynomials for arbitrary values of the argument, the term "Bernoulli polynomials" was introduced by J. L. Raabe (1851). The Bernoulli numbers and polynomials are well studied and find applications in fields of pure and applied mathematics. Various variants of generalization of the Bernoulli numbers and polynomials can be found in [5–11]. A generalization to several variables has been considered in [12]; in this paper definitions of the Bernoulli numbers and polynomials associated with rational lattice cones were given and multidimensional analogs of their basic properties were proved. This paper is devoted to generalization of these results to the case of hypercomplex variables. The Clifford algebra in hypercomplex function theory (HFT) was first used by R. Fueter [1] in
the beginning of the last century. A systematic study of this topic can be found in [2–4]. Also, the papers [15–18] with further advancement of the Clifford analysis should be noted. The notion of the Bernoulli numbers and polynomials in this framework were given and studied in [13, 14].
In this paper we give a more genral notion of Bernoulli polynomials than in [13, 14], namely, in the spirit of [12] we define polynomials in hypercomplex variables associated with a matrix of integers. In the second section of the paper we formulate and prove basic properties of such
polynomials.