• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.243-257
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• 2022

In this paper, we present and prove an new theorem on symmetric functions. By using this theorem, we derive some new generating functions of the products of (p, q)-Fibonacci numbers, (p, q)-Lucas numbers, (p, q)-Pell numbers, (p, q)-Pell Lucas numbers, (p, q)-Jacobsthal numbers and (p, q)-Jacobsthal Lucas numbers with Chebyshev polynomials of the first kind.

• Kyungpook Mathematical Journal
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• v.61 no.3
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• pp.473-486
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• 2021

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.63-73
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• 2023

In this paper, we determine all repdigits, which are difference of two Pell and Pell-Lucas numbers. It is shown that the largest repdigit which is difference of two Pell numbers is 99 = 169 - 70 = P₇ - P₆ and the largest repdigit which is difference of two Pell-Lucas numbers is 444 = 478 - 34 = Q₇ - Q₄.

• Honam Mathematical Journal
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• v.40 no.1
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• pp.199-210
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• 2018

In this paper we obtain some formulae for several sums of generalized Fibonacci numbers $U_n$ and generalized Lucas numbers $V_n$ and their dual forms $G_n$ and $H_n$ by using extensions of an interesting identity by A. R. Amini for Fibonacci numbers to these four kinds of generalizations and their first and second derivatives.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.539-550
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• 2023

In this paper, we define the generalized k-balancing numbers {B^(k)_n} and k-Lucas balancing numbers {C^(k)_n} and associated polynomials, where n is of the form sk+r, 0 ≤ r < k. We give several formulas for these new sequences in terms of classic balancing and Lucas balancing numbers and study their properties. Moreover, we give a Binet style formula, Cassini's identity, and binomial sums of these sequences.

• Honam Mathematical Journal
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• v.42 no.1
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• pp.49-62
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• 2020

In this paper, we consider interesting binomial and alternating binomial triple sums evaluated in multiplication forms in terms of the Fibonacci and Lucas numbers.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.89-96
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• 2023

Edgar obtained an identity between Fibonacci and Lucas numbers which generalizes previous identities of Benjamin-Quinn and Marques. Recently, Dafnis provided an identity similar to Edgar's. In the present article we give some generalizations of Edgar's and Dafnis's identities.

• Honam Mathematical Journal
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• v.45 no.4
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• pp.572-584
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• 2023

In this study, we search for Pell and Pell-Lucas numbers, which are concatenations of two repdigits and find these numbers to be only 12, 29, 70 and 14, 34, 82, respectively. We use Baker's Theory and Baker-Davenport basis reduction method while finding the solutions.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.695-704
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• 2023

Area problems for triangles and polygons whose vertices have Fibonacci numbers on a plane were presented by A. Shriki, O. Liba, and S. Edwards et al. In 2017, V. P. Johnson and C. K. Cook addressed problems of the areas of triangles and polygons whose vertices have various sequences. This paper examines the conditions of triangles and polygons whose vertices have Lucas sequences and presents a formula for their areas.

• Honam Mathematical Journal
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• v.42 no.2
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• pp.235-249
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• 2020

Let Q and R be the well-known matrices associated with Fibonacci and Lucas numbers, and k, m, and n be any integers. It is mainly established all solutions of the matrix equations c₁Qⁿ + c₂Q^m = Q^k, c₁Qⁿ + c₂Q^m = RQ^k, and c₁Qⁿ + c₂RQ^m = Q^k with unknowns c₁, c₂ ∈ ℂ*. Moreover, using the obtained results, it is presented many identities, some of them are available in the literature, and the others are new, related to the Fibonacci and Lucas numbers.