• Kyungpook Mathematical Journal
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• v.64 no.3
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• pp.395-405
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• 2024

In this paper we define a matrix sequence called the Pell matrix sequence whose elements consist of Pell numbers. Using a positive parameter k, we generalize the Pell matrix sequence to a k-Pell matrix sequence and using two parameters s, t we generalize them to (s, t)-Pell matrix sequences. We give the basic properties of these matrix sequences. Then, using these properties we obtain exponential representations of the Pell matrix sequence and its generalizations in different ways.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.1
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• pp.97-111
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• 2019

The purpose of work is to study the sums of k distanced numbers in Pell sequence {$P_i$} as well as in the generalized ${\lambda}-Pell$ sequence {$P_{{\lambda},i}$}.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.23-34
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• 2019

In this paper, we first define generalizations of Pell and Pell-Lucas sequences by the recurrence relations $$p_n=2ap_{n-1}+(b-a^2)p_{n-2}\;and\;q_n=2aq_{n-1}+(b-a^2)q_{n-2}$$ with initial conditions $p_0=0$, $p_1=1$, and $p_0=2$, $p_1=2a$, respectively. We give generating functions and Binet's formulas for these sequences. Also, we obtain some identities of these sequences.

• Honam Mathematical Journal
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• v.41 no.1
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• pp.197-206
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• 2019

We consider the extended generalized Hecke groups ${\bar{H}}_{3,q}$ generated by $X(z)=-(z-1)^{-1}$, $Y(z)=-(z+{\lambda}_q)^{-1}$ with ${\lambda}_q=2\;cos({\frac{\pi}{q}})$ where $q{\geq}3$ an integer. In this work, we study the generalized Pell sequences in ${\bar{H}}_{3,q}$. Also, we show that the entries of the matrix representation of each element in the extended generalized Hecke Group ${\bar{H}}_{3,3}$ can be written by using Pell, Pell-Lucas and modified-Pell numbers.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.535-547
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• 2019

In this paper, by using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and $Peth{\ddot{o}}$, we find all generalized Fibonacci numbers which are Pell numbers. This paper continues a previous work that searched for Pell numbers in the Fibonacci sequence.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.327-334
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• 2021

In this paper, we find all the prime numbers p that satisfy the following statement. If a positive integer k is a divisor of p - 1, then there is a sequence consisting of all k-th power residues modulo p, satisfying the recurrence equation of the Fibonacci sequence modulo p.