• Journal of applied mathematics & informatics
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• 제40권1_2호
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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.

• Korean Journal of Mathematics
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• 제31권1호
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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.

• 대한수학회보
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• 제52권5호
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• pp.1467-1480
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• 2015

Generalized Fibonacci and Lucas sequences ($U_n$) and ($V_n$) are defined by the recurrence relations $U_{n+1}=PU_n+QU_{n-1}$ and $V_{n+1}=PV_n+QV_{n-1}$, $n{\geq}1$, with initial conditions $U_0=0$, $U_1=1$ and $V_0=2$, $V_1=P$. This paper deals with Fibonacci and Lucas numbers of the form $U_n$(P, Q) and $V_n$(P, Q) with the special consideration that $P{\geq}3$ is odd and Q = -1. Under these consideration, we solve the equations $V_n=5kx^2$, $V_n=7kx^2$, $V_n=5kx^2{\pm}1$, and $V_n=7kx^2{\pm}1$ when $k{\mid}P$ with k > 1. Moreover, we solve the equations $V_n=5x^2{\pm}1$ and $V_n=7x^2{\pm}1$.

• 호남수학학술지
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• 제44권4호
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• pp.473-484
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• 2022

In this study, some new relations between generalized Fibonacci numbers and matrices are given. The work is designed in three stages: Firstly, it is obtained a relation between generalized Fibonacci numbers and integer powers of the matrices X satisfying the relation X² = pX +qI, and also, many results are derived from obtained relation. Then, it is established more general relation between generalized Fibonacci numbers and the square matrices X satisfying the condition X² = V_nX - (-q)ⁿI. Finally, some applications and numerical examples related to the obtained results are given.