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Lucas-type associated polynomials

Guettai Ghania, Laissaoui Diffalah, Rahmani Mourad

Mathematica Applicanda

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2023

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Vol. 51, no. 2

291--304

EN

In this paper, we define a new type of Lucas polynomials known as Lucas-type associated polynomials and investigate their fundamental properties and identities. An interesting formula for Lucas-type associated polynomials can be derived using Leibniz's rule for derivatives, defined by Rodrigue's Lucas-type formula. Additionally, we establish an integral connection between Lucas-type associated polynomials and associated Fibonacci polynomials.

PL

W tym artykule definiujemy nowy rodzaj wielomianów Lucasa, znanych jako wielomiany powiązane typu Lucasa, i badamy ich podstawowe właściwości i tożsamości. Ciekawy wzór na wielomiany powiązane typu Lucasa można uzyskać, używając reguły Leibniza dla pochodnych, zdefiniowanej przez wzór Rodrigue’a dla wielomianów powiązanych typu Lucasa. Dodatkowo ustanawiamy związek całkowy między wielomianami powiązanymi typu Lucasa a wielomianami Fibonacciego

2

A note on the convergence of Phillips operators by the sequence of functions via q-calculus

Kiliçman Adem, Ayman-Mursaleen Mohammad, Nasiruzzaman Md.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

615--633

EN

The basic aim of this study is to include nonnegative real parameters to allow for approximation findings of the Stancu variant of Phillips operators. We concentrate on the uniform modulus of smoothness in a simple manner before moving on to the approximation in weighted Korovkin’s space. Our study’s goals and outcomes are to fully develop the uniformly approximated findings of Phillips operators. We determine the order of convergence in terms of Lipschitz maximal function and Peetre’s K-functional. In addition, the Voronovskaja-type theorem is also proved.

3

Moments of the weighted Cantor measures

Harding Steven N., Riasanovsky Alexander W. N.

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

256--273

EN

Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying [wzór] where φn : x ↦ (x+n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L2μα[0,1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xmdμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the firstmmoments within uniform error ε in O((loglog(1/ε))·m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2,1/2].

4

Investigation of G-network with signals at transient behavior

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

75--86

EN

G-queueing network with positive messages and signals at transient behavior is considered. A system of difference-differential equations for the state probabilities of the network is obtained. To find them and the average characteristics of the network a technique was applied based on the use of apparatus of multivariate generating functions. An expression for the generating function was obtained. An example is calculated.

5

Generalized elliptic-type integrals and generating functions

Chaurasia V. B. L., Singh Y.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

310--323

EN

On account of analytical importance or application in certain problems in radiation physics and nuclear technology, several interesting families of elliptic-type integrals were recently studied by many authors. The aim and objective of present paper is to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. In addition, to deriving known and various new elliptic-type integrals and their generalizations, these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.

6

Zastosowanie obliczeń symbolicznych do generowania rozwiązania równania falowego

Ciałkowski M., Jarosławski M.

Zeszyty Naukowe Politechniki Poznańskiej. Maszyny Robocze i Transport

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2003

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Nr 56

115-140

PL

Na podstawie metody rozdzielania zmiennych przedstawiono sposób generowania kolejnych funkcji tożsamościowo spełniających zadane równanie różniczkowe (funkcje Trefftza). Podstawą do wyznaczania tych funkcji jest funkcja generująca zależna od parametru (lub parametrów). Kolejne pochodne funkcji generującej względem parametru wyznaczają kolejne funkcje spełniające zadane równanie różniczkowe. Za pomocą operacji symbolicznych wyznacza się w postaci jawnej poszukiwane funkcje i ich pochodne. Funkcje te mogą być zastosowane jako funkcje bazowe w metodzie elementu skończonego.

EN

The following article presents the description and sphere of activity of a nozzle-flapper system which works as the first part of an amplifier in the hydraulic servovalve. Moreover, on the basis of static characteristics of the amplifier the description and procedures of the parameters choice are described. The methodology was verified by the comparison of results with magnitudes of real parameters of the MOOG 76-225 model servovalve.