A bijective proof of Riordan's theorem on powers of Fibonacci numbers

Let Fk(x) = n=0 Fk n xn . Using the interpretation of Fibonacci numbers in the terms of Morse codes, we give a bijective proof of Riordan's formula (1 - Lkx + (-1)k x2 )Fk(x) = 1 + kx k/2 j=1 (-1)j j akjFk-2j((-1)j x), where Lk = Fk + Fk-2, and akj is defined by means of (1 - x - x2 )-j = k=2j akjxk-2j . The Fibonacci numbers Fn may be defined by F0 = 1, F1 = 1, Fn = Fn-1 + Fn-2, n 2. Their generating function is F1(x) = n=0 Fnxn = (1 - x - x2 )-1 . More generally we may put Fk(x) = n=0 Fk n xn . Riordan [2] has proved that Fk(x) satisfies the following recurrence relation: (1 - Lkx + (-1)k x2 )Fk(x) = 1 + kx k/2 j=1 (-1)j j akjFk-2j((-1)j x), (1) 1 

• Type: Journal
• Year: 1999
• Where: DM
• Authors: Andrej Dujella
• Keywords: Bijective Proof, DM 1999, Fibonacci Numbers, Morse Codes