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1

On h(x) Lucas polynomials with application to coding

Prasad Bandhu

Control and Cybernetics

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2019

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Vol. 48, No. 4

557--573

EN

In this paper, we study the h(x) Lucas polynomials of order m and the Un matrix, whose elements are Lucas polynomials, h(x)(> 0) being a polynomial with real coefficients. We also establish the relations among the code matrix elements.

2

Connections between the primitive 5-th roots of unity and Fibonacci numbers - part II

Wituła R., Jama D.

Zeszyty Naukowe. Matematyka - Fizyka / Politechnika Śląska

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2010

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z. 92

61-73

EN

This paper constitute a continuation of previous paper of the authors. Also in here, many of the classical and relatively recently discovered identities for Fibonacci and Lucas numbers are proven. It seems, that the proposed approach to the problem of proving identities for those numbers is absolutely original.

PL

Praca jest kontynuacją, wcześniejszego artykulu au-torow. Dotyczy nowej, oryginalnej metody dowodzenia różnych tożsamości i zależności dla liczb Fibonacciego i Lucasa.

3

Connections between the primitive 5-th roots of unity and Fibonacci numbers - part I

Wituła R., Jama D.

Zeszyty Naukowe. Matematyka - Fizyka / Politechnika Śląska

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2010

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z. 92

43-60

EN

In this paper a new method of investigating the identities for Fibonacci and Lucas numbers is presented. This method is based on some fundamental identities for powers of the number called the golden ratio and the conjugate number to it. Those identities give also possibility to connect in the interesting way Fibonacci and Lucas numbers with many important sequences of numbers, like Bernoulli numbers, binomial coefficients or Fibonacci numbers.

PL

W artykule przedstawiono nową, metodę dowodzenia tożsamosci dla liczb Fibonacciego i Lucasa. Bazuje ona na pewnych fundamentalnych tożsamościach dla potęg liczby zwanej złotą, proporcją oraz liczby do niej sprzężonej. Tożsamości te pozwalają także ciekawie powiązać liczby Fibonacciego i Lucasa z wieloma ważnymi ciągami liczbowymi, takimi jak: liczby Bernoullego, współczynniki dwumianowe czy liczby 5-Fibonacciego.

4

Sums of products of generalized Fibonacci and Lucas numbers

Cerin Z.

Demonstratio Mathematica

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2009

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Vol. 42, nr 2

247-258

EN

In this paper we obtain explicit formulae for sums of products of a fixed number of consecutive generalized Fibonacci and Lucas numbers. These formulae are related to the recent work of Belbachir and Bencherif. We eliminate all restrictions about the initial values and the form of the recurrence relation. In fact, we consider six different groups of three sums that include alternating sums and sums in which terms are multiplied by binomial coefficients and by natural numbers. The proofs are direct and use the formula for the sum of the geometric series.