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Let Tm be a Tribonacci sequence, and let the k-Pell sequence be a generalization of the Pell sequence for k ≥ 2. The first k terms are 0, 0,..., 0, 1, and each term after the forewords is defined by linear recurrence P(k) n = 2P(k) n-1 + P(k) n-2 +... + P(k) n-k. We study the solution of the Diophantine equation P(k) n = Tm for the positive integer (n, k, m) with k ≥ 2. We use the lower bound for linear forms in logarithms of algebraic numbers with the theory of the continued fraction. [ABSTRACT FROM AUTHOR]

Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,..., 0, 1 and each term afterwards is given by the linear recurrence p n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262]. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 . A generalization of the well-known Pell sequence is the k-Pell sequence. For this sequence the first k terms are 0 , ... , 0 , 1 and each term afterwards is given by the linear recurrence P n (k) = 2 P n - 1 (k) + P n - 2 (k) + ⋯ + P n - k (k). In this manuscript, our main objective is to find all k-Pell numbers which are product of two repdigits. This generalizes a result of Erduvan and Keskin (Pell and Pell–Lucas numbers as products of two repdigits, submitted) regarding Pell numbers with the above property. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a operator in order to derive some new symmetric properties of (p,q)-modified Pell numbers and we give some new generating functions of the products of (p,q)-modified Pell numbers with k-Fibonacci and k-Lucas numbers, k-Pell and k-Pell Lucas numbers, k-Jacobsthal and k-Jacobsthal Lucas numbers at positive and negative indices. By making use of the operator defined in this paper, we give some new generating functions of the products of (p,q)-modified Pell numbers with kbalancing and k-Lucas-balancing numbers. [ABSTRACT FROM AUTHOR]

Considering the usual k-Pell numbers we define a new sequence which is called Generalized order-m (k-Pell) sequence. Some properties of this sequence and a formula for sums of k-Pell numbers are obtained. A generalized Binet's formula and some identities involving this sequence are provided and also a combinatorial representation of the Generalized order-m (k-Pell) number is deduced. [ABSTRACT FROM AUTHOR]

In this paper, we define the k-Pell, k-Pell-Lucas and Modified k-Pell numbers with arithmetic indexes, study some properties of these sequences of numbers and give their generating functions. [ABSTRACT FROM AUTHOR]

The aim of this work is to present the k -Pell, the k -Pell–Lucas and the Modified k -Pell sedenions and we give some properties involving these sequences, including the Binet-style formulae and the ordinary generating functions. [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce a new sequence of numbers called k-Pell hybrid numbers and the presentation of some algebraic properties involving this sequence. In addition, we present the Binet formula, the generating functions and some identities related with different terms of the sequence. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new operator in order to derive some properties of homogeneous symmetric functions. By making use of the proposed operator, we give some new generating functions for k-Fibonacci numbers, k-Pell numbers and product of sequences and Chebyshev polynomials of second kind. [ABSTRACT FROM AUTHOR]