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Permutations preserving sums of rearranged real series

100%
Wituła R.
Open Mathematics
|
2013
|
tom 11
|
nr 5
956-965
EN
The aim of this paper is to discuss one of the most interesting and unsolved problems of the real series theory: rearrangements that preserve sums of series. Certain hypothesis about combinatorial description of the corresponding permutations is presented and basic algebraic properties of the family $\mathfrak{S}_0 $, introduced by it, are investigated.
2
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Divergent vector sequences {yₙ} with Δyₙ → 0

100%
Wituła R.
Colloquium Mathematicum
|
2007
|
tom 107
|
nr 2
263-266
EN
An approximation property of divergent sequences in normed vector spaces is discussed.
3
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The Riemann theorem and divergent permutations

100%
Wituła R.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 2
275-287
EN
In this paper the fundamental algebraic propeties of convergent and divergent permutations of ℕ are presented. A permutation p of ℕ is said to be divergent if at least one conditionally convergent series $∑ a_n$ of real terms is rearranged by p to a divergent series $∑ a_{p(n)}$. All other permutations of ℕ are called convergent. Some generalizations of the Riemann theorem about the set of limit points of the partial sums of rearrangements of a given conditionally convergent series are also studied.
4
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Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

64%
Wituła R., Słota D.
Open Mathematics
|
2006
|
tom 4
|
nr 3
531-546
EN
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.
5
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Binomials transformation formulae for scaled Fibonacci numbers

51%
Hetmaniok E., Piątek B., Wituła R.
Open Mathematics
|
2017
|
tom 15
|
nr 1
477-485
EN
The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.
6
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Some new facts about group 𝒢 generated by the family of convergent permutations

51%
Wituła R., Hetmaniok E., Słota D.
Open Mathematics
|
2017
|
tom 15
|
nr 1
568-577
EN
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 only from the permutations preserving the sum of series. This result substantially strengthens some old Pleasants’ result.
7
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Jordan numbers, Stirling numbers and sums of powers

45%
Wituła R., Kaczmarek K., Lorenc P., Hetmaniok E., Pleszczyński M.
Discussiones Mathematicae - General Algebra and Applications
|
2014
|
tom 34
|
nr 2
155-166
EN
In the paper a new combinatorical interpretation of the Jordan numbers is presented. Binomial type formulae connecting both kinds of numbers mentioned in the title are given. The decomposition of the product of polynomial of variable n into the sums of kth powers of consecutive integers from 1 to n is also studied.
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