The Symmetries of (𝒌𝒌:𝜶𝜶𝟏𝟏,𝜶𝜶𝟐𝟐,…,𝜶𝜶𝒌𝒌)-step Fibonacci Functions

Abstract

It is well known that the Fibonacci sequence (𝐹𝑛) is denoted by 𝐹0=0, 𝐹1=1 and 𝐹𝑛=𝐹𝑛−1+𝐹𝑛−2, while the Lucas sequence (𝐿𝑛) is denoted by 𝐿0=2, 𝐿1=1 and 𝐿𝑛=𝐿𝑛−1+𝐿𝑛−2. There are several studies showing relations between these two sequences. An interesting generalisation of both the sequences is a Fibonacci function 𝑓:ℝ→ℝ defined by 𝑓(𝑥+2)=𝑓(𝑥+1)+𝑓(𝑥) for any real number 𝑥 (Elmore, 1967). Research about periods of Fibonacci numbers modulo 𝑚 (Jameson, 2018) results in a contribution on the existence of primitive period of a Fibonacci function 𝑓:ℤ→ℤ modulo 𝑚 (Thongngam & Chinram, 2019). Recently, a 𝑘-step Fibonacci function 𝑓:ℤ→ℤ denoted by 𝑓(𝑛+𝑘)=𝑓(𝑛+𝑘−1)+𝑓(𝑛+𝑘−2)+⋯+𝑓(𝑛) for any integer 𝑛 and 𝑘≥2 (which is a generalisation of a Fibonacci function 𝑓:ℤ→ℤ) is introduced and the existence of primitive period of this function modulo 𝑚 is established (Tongron & Kerdmongkon, 2022). In this work, let 𝑘 be an integer ≥2. For nonnegative integers 𝛼1,𝛼2,…,𝛼𝑘 and 𝛼1≠0, a (𝑘:𝛼1,𝛼2,…,𝛼𝑘)-step Fibonacci function 𝑓:ℤ→ℤ is defined by 𝑓(𝑛)=𝑓(𝑛−𝛼1)+𝑓(𝑛−𝛼1−𝛼2)+⋯+𝑓(𝑛−𝛼1−𝛼2−⋯−𝛼𝑘) for any integer 𝑛. In fact, a 𝑘-step Fibonacci function is a special case of a (𝑘:𝛼1,𝛼2,…,𝛼𝑘)-step Fibonacci function. We present the existence of primitive period of this function modulo 𝑚 and show that certain (𝑘:𝛼1,𝛼2,…,𝛼𝑘)-step Fibonacci functions are symmetric-like.