In this paper, we introduce a new operator in order to derive some properties of homogeneous symmetric functions. By making use of the proposed operator, we give some new generating functions for Mersenne numbers, Mersenne numbers and product of sequences and Chebychev polynomials of second kind.
In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers through the use of Binet’s formula.
In this paper, we will recover the generating functions of generalized polynomials of second order. The technic used her is based on the theory of the so called symmetric functions.
Let F, N, A and N2 denote the properties of being finite, nilpotent, abelian and nilpotent of classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups. We show that the property FC is closed under finite extensions, and extend this result to finitely generated NF-groups. Secondly we prove that a finitely generated NF-group G is in the class ((FC)F, ∞) if and only if G is an FA-group. Finally we prove that a finitely generated NF-group in the class ((FC)F, ∞)* is an FN2-group. Moreover, G/Z2(G) is finite.
In this paper, we define generalized Gaussian Padovan numbers, generalized Gaussian Padovan polynomials and generalized trivariate Fibonacci polynomials, we present the new generating functions of these generalizations, and as special cases, we investigate Gaussian Padovan numbers and polynomials, Gaussian Pell Padovan numbers and polynomials and trivariate Fibonacci and Lucas polynomials with their generating functions. Moreover, we give the new generating functions of some generalized Vieta polynomials.
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