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1
Content available Two-parametric quasi-Fibonacci numbers
Smoleń B., Wituła R.
Silesian Journal of Pure and Applied Mathematics
|
2017
|
Vol. 7, iss. 1
99--121
EN
This paper is devoted to the discussion on the two parametric quasi–Fibonacci numbers. The fundamental recurrence and reduction formulae for arguments and indices of these quasi–Fibonacci numbers are presented here. The matrix representations of the considered numbers are described and their applications are indicated. Moreover, a number of connections of the two parametric quasi-Fibonacci numbers with the sequences collected in the OEIS encyclopaedia are noted. Despite quite large volume of this elaboration, the Authors believe that this is just some kind of announcement, or an introduction to a definitely larger and detailed discussion including, above all, the applications of the investigated here numbers.
2
Content available Variations on the arbelos
Różański M., Samulewicz A., Szweda M., Wituła R.
Journal of Applied Mathematics and Computational Mechanics
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2017
|
Vol. 16, nr 2
123--133
EN
We recall the ancient notion of arbelos and introduce a number of concepts generalizing it. We follow the ideas presented by J. Sondow in his article on parbelos, the parabolic analogue of the classic arbelos. Our concepts concern the curves constructed of arcs which resemble each other and surfaces obtained in a similar way. We pay special attention to ellarbelos, the curves built of semi-ellipses, because of their possible application in engineering, e.g. in determining the static moments of arc rod constructions or in problems of structural stability and durability of constructions.
3
Content available On values of the psi function
Adam M., Piątek B., Pleszczyński M., Smoleń B., Wituła R.
Journal of Applied Mathematics and Computational Mechanics
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2017
|
Vol. 16, nr 1
7--18
EN
In the present paper, values of the psi function for many arguments connected with the golden ratio and Fibonacci numbers are determined or given in alternative form. Moreover, some integral representation of the psi function is found. This is a potential calculation base of values of the psi function for powers of argument. We also note that this integral representation gives better numerical estimation of values of the psi function than the respective Legendre’s integral formula.
4
Content available Certain inequalities connected with the golden ratio and the Fibonacci numbers
Adam M., Piątek B., Pleszczyński M., Smoleń B., Wituła R.
Journal of Applied Mathematics and Computational Mechanics
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2016
|
Vol. 15, nr 1
5--15
EN
In the present paper we give some condensation type inequalities connected with Fibonacci numbers. Certain analytic type inequalities related to the golden ratio are also presented. All results are new and seem to be an original and attractive subject also for future research.
5
Content available Orbits of the Kaprekar’s transformations – some introductory facts
Hanslik H., Hetmaniok E., Sobstyl I., Pleszczyński M., Wituła R.
Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska
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2015
|
z. 5
5--34
EN
Presented paper, above all, completes two other papers, made previously by the authors and cited in References, concerning the orbits of the Kaprekar’s transformations. In the current paper many detailed facts for five initial Kaprekar’s transformations (from T2 to T6) are described. There are introduced some new concepts and there is shown how important is the observation of numerical results giving the motivation for theoretical discussion on the Kaprekar’s transformations. Moreover, the paper includes the interesting and original results concerning the fixed points and the 2-element orbits of the Kaprekar’s transformations. All these results encourage to continue the discussion. The paper contains also quite large survey section devoted to the generalizations and modifications of the Kaprekar’s transformations. Furthermore, some pieces of information from OEIS by N.J.A. Sloane connected with the orbits of the Kaprekar’s transformations are presented here.
PL
Prezentowany artykuł uzupełnia dwie inne prace autorów, cytowane w dołączonej tu literaturze, dotyczące orbit transformacji Kaprekara. W pracy przedstawiono wiele szczegółowych faktów dla pierwszych pięciu transformacji Kaprekara (od T2 do T6). Wprowadzono nowe pojęcia i pokazano, jak istotna jest analiza wyników numerycznych jako motywacja do dyskusji teoretycznej transformacji Kaprekara. Ponadto w artykule zamieszczono interesujące i oryginalne wyniki dotyczące punktów stałych i 2-elementowych orbit transformacji Kaprekara. Wszystkie te wyniki wręcz zachęcają do dalszej dyskusji. Artykuł zawiera również obszerny rozdział dotyczący uogólnień i modyfikacji transformacji Kaprekara. Wreszcie mamy tu zamieszczoną garść informacji z OEIS, ufundowanej przez N.J.A. Sloane’a, związanych z orbitami transformacji Kaprekara.
6
Content available On the limits of quotients of polynomials in two variables
Wituła R., Hetmaniok E., Wróbel A., Matlak J.
Journal of Applied Mathematics and Computational Mechanics
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2015
|
Vol. 14, nr 1
121--132
EN
The aim of this paper is to discuss different types of decompositions and factorizations concerning a few families of symmetric polynomials in two variables including Ma polynomials, classic Cauchy polynomials, Ferrers-Jackson polynomials and some elementary polynomials as well. Application of the discussed decompositions and factorizations for determining the limits of quotients of the respective polynomials in two variables is presented here and some general theorems on these limits are also proven in this elaboration.
7
Content available An analytical method for solving the two-phase inverse Stefan problem
Hetmaniok E., Słota D., Wituła R., Zielonka A.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2015
|
Vol. 63, nr 3
583--590
EN
In the paper we present an application of the homotopy analysis method for solving the two-phase inverse Stefan problem. In the proposed approach a series is created, having elements which satisfy some differential equation following from the investigated problem. We reveal, in the paper, that if this series is convergent then its sum determines the solution of the original equation. A sufficient condition for this convergence is formulated. Moreover, the estimation of the error of the approximate solution, obtained by taking the partial sum of the considered series, is given. Additionally, we present an example illustrating an application of the described method.
8
Content available Sums of the rational powers of roots of cubic polynomials
Wituła R., Lorenc P., Różański M., Szweda M.
Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska
|
2014
|
z. 4
17--34
PL
W artykule przedstawiono elementarną metodę generowania równości i tożsamości trygonometrycznych. Między innymi wyprowadzono równości związane z pierwiastkami wielomianu Perrina oraz otrzymano uogólnienia znanych równości Ramanujana.
EN
In this paper a completely elementary method of generating the trigonometric equalities and identities is presented. Among them, the equalities connected with roots of Perrin’s polynomial and the generalizations of known Ramanujan’s equalities are proven.
9
Content available Solution of some Kronheimer’s problem on summable sequences
Lorenc P., Różański M., Szweda M., Wituła R.
Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska
|
2014
|
z. 4
35--40
EN
In the present paper we give the solution of E. Kronheimer’s problem (problem A6516 in Amer.Math.Month.), alternative to three other solutions included in paper [1].
PL
W artykule przedstawiono rozwiązanie problemu E. Kronheimera (problem A6515 z Amer. Math. Monthly) alternatywne do trzech innych rozwiązań tego problemu zawartych w pracy [1].
10
Content available Differentiation and integration by using matrix inversion
Matlak D., Matlak J., Słota D., Wituła R.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
Vol. 13, nr 2
63--71
EN
In the paper certain examples of applications of the matrix inverses for generating and calculating the integrals are presented.
11
Content available A geometric property of the roots of Chebyshev polynomials
Wituła R., Hetmaniok E., Słota D.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
Vol. 13, nr 1
143--148
EN
In this text a new property of geometric nature of the Chebyshev polynomials is given. It is proven that the lengths of diagonals of a regular n-gon with the side of length equal to one are the sums of positive roots of the respective renormalized Chebyshev polynomials of one from among four types. Some new special decompositions of differences of values of the Chebyshev polynomials are also presented.
12
On series whose rearrangements possess discrete sets of limit points
Wituła R., Hetmaniok E., Kaczmarek K.
Journal of Applied Analysis
|
2014
|
Vol. 20, nr 1
93--96
EN
Aim of this short paper is to construct in any infinite-dimensional Hilbert space a series with terms tending to zero such that some of its rearrangements possess the discrete set of limit points.
13
Content available remote Certain multiplier version of the Riemann derangement theorem
Wituła R.
Demonstratio Mathematica
|
2014
|
Vol. 47, nr 1
125--129
EN
Aim of this paper is to consider a problem formulated in [6]. Namely, it has been proven that for any sequences {xn} (…) , for every interval (…) , there exist a nondecreasing sequence (…) of positive integers and a sequence (…) signs such that the set of limit points of the series (…) is equal to [a, b].
14
Content available An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit
Hetmaniok E., Słota D., Trawiński T., Wituła R.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2014
|
Vol. 62, nr 3
413--421
EN
In this paper an application of the homotopy perturbation method for solving the general linear integral equations of the second kind is discussed. It is shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in the homotopy perturbation method is convergent. The error of approximate solution, received by taking only the partial sum of the series, is also estimated. Moreover, there is presented an example of applying the method for approximate solution of an equation which has a practical application for charge calculation in supply circuit of the flash lamps used in cameras.
15
Content available Some intriguing limits - continuation
Lorenc P., Wituła R.
Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska
|
2013
|
z. 3
17--24
EN
Wituła and Słota in [College Math. J. 42 (2011), 328] proposed a way of proving the relation (1) given below which appeared to be a genuine result. Authors of the present paper, inspired by the form of this limit, try to find some generalizations of this one, also in the context of some special functions (e.g. the gamma function, the generalized Laguerre polynomials).
PL
Wituła i Słota w notce [College Math. J. 42 (2011), 328] zaproponowali udowodnienie relacji (1), podanej poniżej, która wydaje się bardzo ciekawą zależnością. Autorzy niniejszego artykułu, zainspirowani postacią tej granicy, próbują znaleźć różne jej uogólnienia także w kontekście pewnych funkcji specjalnych (np. funkcji gamma, uogólnionych wielomianów Laguerre’a).
16
Content available Series of iterated logarithms
Różański M., Szweda M., Wituła R.
Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska
|
2013
|
z. 3
37--46
EN
In this paper certain properties of the series of iterated logarithms are discussed in more general context.
PL
W artykule pewne własności szeregów iterowanych logarytmów są rozważane w znacznie ogólniejszym kontekście.
17
Content available Darboux property of the nonatomic σ-additive positive and finite dimensional vector measures
Lorenc P., Wituła R.
Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska
|
2013
|
z. 3
25--36
EN
In the paper some facts connected with Darboux property of the positive measures and the finite dimensional vector measures are discussed.
PL
W artykule omawiane są pewne fakty związane z własnością Darboux miar nieujemnych i miar wektorowych skończenie wymiarowych.
18
Content available Some generalizations of Gregory's power series and their applications
Gawrońska N., Słota D., Wituła R., Zielonka A.
Journal of Applied Mathematics and Computational Mechanics
|
2013
|
Vol. 12, nr 3
79--91
EN
The series representing the generalizations of classical James Gregory's series are discussed in this paper. Formulae describing sums of these series are found. A number of applications of obtained formulae are also presented, among others, in receiving the generalizations of Gregory-Leibniz-Nilakantha formula. Moreover, the sums of series of differences of odd harmonic numbers are generated.
19
Content available On similarities between exponential polynomials and hermite polynomials
Hetmaniok E., Pleszczyński M., Słota D., Wituła R.
Journal of Applied Mathematics and Computational Mechanics
|
2013
|
Vol. 12, nr 3
93--104
EN
The aim of this paper is to introduce and compare some fundamental analytical properties of the title polynomials. Many similarities between them are emphasized in the paper. Moreover, the authors present many isolated results, new proofs and identities.
20
Content available remote Binomials transformation formulae of scaled Lucas numbers
Wituła R.
Demonstratio Mathematica
|
2013
|
Vol. 46, nr 1
15--27
EN
The current paper represents a suplement for papers [7] and [8]. Many of the new summation formulae connecting Lucas numbers with binomials are presented here. All these relations are obtained by using definition and simple properties of the so called δ-Lucas numbers.
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