Generalized Euler polynomials α n (x) = (1 − x) n+1 ∞ m=0 p n (m) x m , where p n (x) is the polynomial of degree n, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan arrays. Generalized Narayana polynomialsx m are the numerator polynomials of the generating functions of diagonals of the exponential Riordan arrays. In present paper we consider the constructive relationship between these two types of numerator polynomials.
Set of generalized Pascal matrices whose elements are generalized binomial coefficients is considered as an integral object. The special system of generalized Pascal matrices, based on which we are building fractal generalized Pascal matrices, is introduced. Pascal matrix (Pascal triangle) is the Hadamard product of the fractal generalized Pascal matrices whose elements equal to p k , where p is a fixed prime number, k = 0, 1, 2, . . . .The concept of zero generalized Pascal matrices, an example of which is the Pascal triangle modulo 2, arise in connection with the system of matrices introduced.
Ordinary algebra of formal power series in one variable is convenient to study by means of the algebra of Riordan matrices and the Riordan group. In this paper we consider algebra of formal power series without constant term, isomorphic to the algebra of formal Dirichlet series. To study it, we introduce matrices, similar to the Riordan matrices. As a result, some analogies between two algebras becomes visible. For example, the Bell polynomials (polynomials of partitions of number n into m parts) play a certain role in the ordinary algebra. Similar polynomials (polynomials of decompositions of number n into m factors) play a similar role in the considered algebra. Analog of the Lagrange series for the considered algebra is also exists. In connection with this analogy, we introduce matrix group, similar to the Riordan group and called the Riordan-Dirichlet group. As an example, we consider analog of the Abel's identities for this group.
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