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Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

Roman Wituła;  Damian Słota — 2006

Open Mathematics

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x 2n + y 2n , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.

Some new facts about group 𝒢 generated by the family of convergent permutations

Roman Wituła;  Edyta Hetmaniok;  Damian Słota — 2017

Open Mathematics

The aim of this paper is to present some new and essential facts about group 𝒢 generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to 𝒢. Additionally, we show that there exists a family G (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than 𝒢 and is composed...

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