Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 27, 2021, No. 1, 198–207 DOI: 10.7546/nntdm.2021.27.1.198-207 Catalan transform of the k-Pell, k-Pell–Lucas and modified k-Pell sequence Merve Taştan1 and Engin Özkan2 1 Graduate School of Natural and Applied Sciences Erzincan Binali Yıldırım University 24100 Erzincan, Turkey e-mail: [email protected] 2 Department of Mathematics, Faculty of Arts and Sciences Erzincan Binali Yıldırım University 24100 Erzincan, Turkey e-mail: [email protected] Received: 15 May 2020 Revised: 11 December 2020 Accepted: 27 January 2021 Abstract: In this study, we present the Catalan transforms of the k-Pell sequence, the k-Pell–Lucas sequence and the Modified k-Pell sequence and examine the properties of the sequences. Then we apply the Hankel transform to the Catalan transforms of the k-Pell sequence, the Catalan transform of the k-Pell–Lucas sequence and the Catalan transform of the Modified k-Pell sequence. Also, we obtain the generating functions of the Catalan transform of the k-Pell sequence, k-Pell–Lucas sequence and Modified k-Pell sequence. Furthermore, we acquire an interesting characteristic related to the determinant of the Hankel transform of the sequences. Keywords: k-Pell numbers, k-Pell–Lucas, Catalan numbers, Catalan transform, Hankel transform. 2010 Mathematics Subject Classification: 11B39, 11B83, 11C20. 1 Introduction The known Pell numbers have some applications in many branches of mathematics such as group theory, calculus, applied mathematics, linear algebra, etc. [2, 3, 4, 6, 8, 16]. There is an extensive work in the literature concerning Fibonacci-type sequences and their applications in modern science (see e.g. [10, 11, 12] and the references therein). 198 There exist generalizations of the Pell numbers. This paper is an extension of the work of Falcon [7], where the author gave an application of the Catalan transform to the k-Fibonacci sequences. Özkan et al. defined a new family of Catalan k-Lucas numbers and gave some properties about the family of these numbers [14]. There are some works on polynomials of the families of k-Fibonacci numbers and k-Lucas numbers [13]. Rajkovic et al. gave some properties of the Hankel transformation of a sequence [15]. One of the latest works in this area is [17] where it is presented the Catalan transforms of the k-Jacobsthal sequence and examined the properties of the sequence. They also applied the Hankel transform to the Catalan transform of the k-Jacobsthal sequence. In this work, we consider the Catalan transform to the k-Pell, k-Pell–Lucas and Modified k-Pell numbers and present an application of the Hankel transform to the Catalan transform of these sequences. The paper is prepared as follows. In the following section, we recall some fundamental definitions of k-Pell numbers, k-Pell–Lucas numbers and Modified k-Pell numbers. In Section 3, the Catalan transforms of k-Pell sequence and k-Pell–Lucas sequence are given. Also, we obtain the generating functions of the Catalan transform of the k-Pell sequence, k-Pell–Lucas sequence and the Modified k-Pell sequence. Finally, we give the Hankel transforms of the new sequence obtained from k-Pell, k-Pell–Lucas and Modified k-Pell sequences. 2 Materials and methods 2.1 k-Pell, k-Pell–Lucas numbers and Modified k-Pell numbers For any positive real number k, the k-Pell sequence, {}Pk, n n , is defined by the recurrence relation Pk, n 12 P k , n kP k , n 1 , n 1, where Pk ,0 0 and Pk ,1 1, [2]. The k-Pell–Lucas sequence, {푄푘,푛}푛∈ℕ, is defined as Qk, n 12 Q k , n kQ k , n 1 , n 1, where Qk,0 2 and Qk ,1 = 2, [3]. The Modified k-Pell sequence, {푞푘,푛}푛∈ℕ, is defined as qk, n 12 q k , n kq k , n 1 , n 1, where qk,0 1 and qk ,1 1, [2]. The Binet formula, respectively, for k-Pell sequence, k-Pell–Lucas sequence and Modified k-Pell sequence is given by: nn nn rr12 nn rr12 Pkn, , Qkn, r 1 r 2 , qkn, , rr12 2 where rk1 11 and rk2 11 , respectively, [2]. Note that, since k 0 , rr122 , rr12 k and r12 r 21 k . 199 2.2. Catalan numbers The Catalan numbers are defined by 1 2n Cn n 1n in [1]. The latter can be written as 2!n C . n nn1 ! ! Also, one can obtain the recurrence relation for Cn from Cn1 2 2n 1 Cnn 2 in [5]. The first few Catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, … [18]. 3 Main results 3.1 Catalan transform of the k-Pell sequence Now, following Falcon in [7], we introduce the Catalan transform of the k-Pell sequence {푃푘,푛} as n i 2ni CPk,, n P k i i0 2ni ni for n 1 with CPk,0 0. We can give some of them as follow: 푖 퐶푃 = ∑1 (2−푖)푃 = 1, 푘,1 1 2−푖 1−푖 푘,푖 푖 퐶푃 = ∑2 (4−푖)푃 = 3, 푘,2 1 4−푖 2−푖 푘,푖 푖 퐶푃 = ∑3 (6−푖)푃 = 10 + 푘, 푘,3 1 6−푖 3−푖 푘,푖 퐶푃푘,4 = 35 + 7푘, 2 퐶푃푘,5 = 126 + 37푘 + 푘 , 2 퐶푃푘,6 = 462 + 176푘 + 11푘 , 2 3 퐶푃푘,7 = 1716 + 794푘 + 80푘 + 푘 . We give the coefficient of 푘 in Table 1 for 0 ≤ ≤ 푛. The following equation can be written as a product of two matrices 퐶. 푃 푃 푘,1 1 푘,1 퐶. 푃 푃 푘,2 1 1 푘,2 퐶. 푃푘,3 푃푘,3 2 2 1 퐶. 푃푘,4 = 5 5 3 1 푃푘,4 . 14 14 9 4 1 퐶. 푃푘,5 푃푘,5 퐶. 푃 42 42 28 14 5 1 푃 푘,6 [ ] 푘,6 [ ⋮ ] ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ [ ⋮ ] 200 푖−1 The entries of the matrix 퐶 verify the recurrence relation 퐶푖,푗 = ∑푟=푗−1 퐶푖−1,푟. For > 1, note that the first and second columns are the Catalan numbers. The matrix 퐶푛,푛−푖 is the Catalan triangle. For 1 ≤ ≤ 푛, the entries of the matrix 퐶푛,푛−푖 verify (2푛 − )! ( + 1) 퐶 = . 푛,푛−푖 (푛 − )! (푛 + 1)! 퐶푃1 1 퐶푃2 3 퐶푃3 1 10 퐶푃4 7 35 퐶푃5 1 37 126 퐶푃6 11 176 462 퐶푃7 1 80 794 1716 … … … … … Table 1. Catalan triangle of the k-Pell sequence We obtain the first few Catalan transform of the k-Pell sequence as follows: 퐶푃1 = {0, 1, 3, 11, 42, 164,649, … }, indexed in OEIS as A143464. 퐶푃2 = {0, 1, 3, 12, 49, 204, 858, … }, 퐶푃3 = {0, 1, 3, 13, 56, 246, 1089, … }, 퐶푃4 = {0, 1, 3, 14, 63, 290, 1342, … }, 퐶푃5 = {0, 1, 3, 15, 70, 336, 1617, … }. 3.2 Catalan transform of the k-Pell–Lucas sequence Similarly, we introduce the Catalan transform of the k-Pell–Lucas sequence {푄푘,푛} as 푛 2푛 − 퐶푄 = ∑ ( ) 푄 푘,푛 2푛 − 푛 − 푘,푖 푖=0 for 푛 ≥ 1 with 퐶푄푘,0 = 0. We can give some of them as follows: 푖 퐶푄 = ∑1 (2−푖)푄 = 2, 푘,1 1 2−푖 1−푖 푘,푖 푖 퐶푄 = ∑2 (4−푖)푄 = 6 + 2푘, 푘,2 1 4−푖 2−푖 푘,푖 푖 퐶푄 = ∑3 (6−푖)푄 = 20 + 10푘, 푘,3 1 6−푖 3−푖 푘,푖 2 퐶푄푘,4 = 70 + 44푘 + 2푘 , 2 퐶푄푘,5 = 252 + 186푘 + 18푘 , 2 3 퐶푄푘,6 = 924 + 772푘 + 114푘 + 2푘 , 2 3 퐶푄푘,7 = 3432 + 3172푘 + 624푘 + 26푘 . We give the coefficient of 푘 in Table 2 for 0 ≤ ≤ 푛. 201 The following equation can be written as the product of two matrices 퐶푄 = 퐶. 푄, where 푡 푡 퐶푄 = (퐶푄푘,푖) and 푄 = (푄푘,푖) . 퐶. 푄 푄 푘,1 1 푘,1 퐶. 푄 푄 푘,2 1 1 푘,2 퐶. 푄푘,3 푄푘,3 2 2 1 퐶. 푄푘,4 = 5 5 3 1 푄푘,4 . 14 14 9 4 1 퐶. 푄푘,5 푄푘,5 퐶. 푄 42 42 28 14 5 1 푄 푘,6 [ ] 푘,6 [ ⋮ ] ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ [ ⋮ ] 퐶푄1 2 퐶푄2 2 6 퐶푄3 10 20 퐶푄4 2 44 70 퐶푄5 18 186 252 퐶푄6 2 114 772 924 퐶푄7 26 624 3172 3432 … … … … … Table 2. Catalan triangle of the k-Pell–Lucas sequence We obtain the first few Catalan transform of the k-Pell–Lucas sequence as follows: 퐶푄1 = {0, 2, 8, 34, 116, 456,1812, … }, 퐶푄2 = {0, 2, 10, 48, 166, 696, 2940, … }, 퐶푄3 = {0, 2, 12, 62, 220, 972,4320, … }, 퐶푄4 = {0, 2, 14, 76, 278, 1284, 5964, … }, 퐶푄5 = {0, 2, 16, 90, 340, 1632, 7884, … }. 3.3 Catalan transform of the Modified k-Pell sequence Now, following Falcon in [7], we introduce the Catalan transform of the Modified k-Pell sequence {푞푘,푛} as 푛 2푛 − 퐶푞 = ∑ ( ) 푞 푘,푛 2푛 − 푛 − 푘,푖 푖=0 for 푛 ≥ 1 with 퐶푞푘,0 = 0. We can give some of them as follows: 푖 퐶푞 = ∑1 (2−푖)푞 = 1, 푘,1 1 2−푖 1−푖 푘,푖 푖 퐶푞 = ∑2 (4−푖)푞 = 3 + 푘, 푘,2 1 4−푖 2−푖 푘,푖 푖 퐶푞 = ∑3 (6−푖)푞 = 10 + 5푘, 푘,3 1 6−푖 3−푖 푘,푖 2 퐶푞푘,4 = 35 + 22푘 + 푘 , 202 2 퐶푞푘,5 = 126 + 93푘 + 9푘 , 2 3 퐶푞푘,6 = 462 + 386푘 + 57푘 + 푘 , 2 3 퐶푞푘,7 = 1716 + 1586푘 + 312푘 + 13푘 .