Abstract
It is well known that the Fibonacci sequence (๐น๐) is denoted by ๐น0=0, ๐น1=1 and ๐น๐=๐น๐โ1+๐น๐โ2, while the Lucas sequence (๐ฟ๐) is denoted by ๐ฟ0=2, ๐ฟ1=1 and ๐ฟ๐=๐ฟ๐โ1+๐ฟ๐โ2. There are several studies showing relations between these two sequences. An interesting generalisation of both the sequences is a Fibonacci function ๐:โโโ defined by ๐(๐ฅ+2)=๐(๐ฅ+1)+๐(๐ฅ) for any real number ๐ฅ (Elmore, 1967). Research about periods of Fibonacci numbers modulo ๐ (Jameson, 2018) results in a contribution on the existence of primitive period of a Fibonacci function ๐:โคโโค modulo ๐ (Thongngam & Chinram, 2019). Recently, a ๐-step Fibonacci function ๐:โคโโค denoted by ๐(๐+๐)=๐(๐+๐โ1)+๐(๐+๐โ2)+โฏ+๐(๐) for any integer ๐ and ๐โฅ2 (which is a generalisation of a Fibonacci function ๐:โคโโค) is introduced and the existence of primitive period of this function modulo ๐ is established (Tongron & Kerdmongkon, 2022). In this work, let ๐ be an integer โฅ2. For nonnegative integers ๐ผ1,๐ผ2,โฆ,๐ผ๐ and ๐ผ1โ 0, a (๐:๐ผ1,๐ผ2,โฆ,๐ผ๐)-step Fibonacci function ๐:โคโโค is defined by ๐(๐)=๐(๐โ๐ผ1)+๐(๐โ๐ผ1โ๐ผ2)+โฏ+๐(๐โ๐ผ1โ๐ผ2โโฏโ๐ผ๐) for any integer ๐. In fact, a ๐-step Fibonacci function is a special case of a (๐:๐ผ1,๐ผ2,โฆ,๐ผ๐)-step Fibonacci function. We present the existence of primitive period of this function modulo ๐ and show that certain (๐:๐ผ1,๐ผ2,โฆ,๐ผ๐)-step Fibonacci functions are symmetric-like.
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