CATALAN TRANSFORM OF THE κ-FIBONACCI SEQUENCE

Abstract

Abstract. In this paper we apply the Catalan transform to the k-Fibo-nacci sequence finding different integer sequences, some of which are in-dexed in OEIS and others not. After we apply the Hankel transform tothe Catalan transform of the k-Fibonacci sequence and obtain an unusualproperty. 1. IntroductionThe classical Fibonacci numbers have been very used in as different sciencesas the biology, demography or economy [7]. Recently they have been appliedeven in the high-energy physics [10, 11, 12, 13]. But there exist generaliza-tions of these numbers given by researches as Horadam [8] and recently bywe ourselves [3, 4, 5]. Now we present this last generalization, so called thek-Fibonacci numbers.1.1. k-Fibonacci numbersFor any positive real number k, the k-Fibonacci sequence, say {F k,n } n∈N , isdefined recurrently by(1) F k,n+1 = k F k,n +F k,n−1 for n ≥ 1with initial conditions F k,0 = 0 and F k,1 = 1.For k = 1, the classical Fibonacci sequence is obtained and for k = 2, thePell sequence appears.The well-known Binet’s formula in the Fibonacci numberstheory [3, 8, 16]allows us to express the k-Fibonacci numbers in function of the roots σ