Convolved Numbers of $k$-sections of the Fibonacci Sequence: Properties, Consequences

TL;DR

This paper introduces a new generalization of Fibonacci numbers through convolutions of k-sections, providing explicit formulas and exploring their properties and connections to Chebyshev polynomials and Lucas numbers.

Contribution

It defines convolutions of k-sections of Fibonacci numbers, derives explicit formulas, and explores their mathematical properties and connections to other special functions.

Findings

Derived an explicit formula for convolutions of k-sections of Fibonacci numbers.

Established a Binet type formula for these convolutions.

Connected these sequences to derivatives of Chebyshev polynomials and Lucas numbers.

Abstract

One possible data encryption scheme is related to stream ciphers, which use a sufficiently long pseudo-random sequence. To increase the cryptographic strength of the cipher, linear shift algorithms (generated by linear recurrent sequences such as the Fibonacci sequence and its generalizations) are additionally used. Two such generalizations are convolved Fibonacci numbers $\{F_n^{(s)}\}_{n=1}^\infty$ and k-sections of the Fibonacci sequence $\{\Phi_{n,k}\}_{n=1}^\infty$ $( \Phi_{n,k}=F_{nk}/F_k).$ This article considers a further generalization of Fibonacci numbers, namely convolutions of k-sections of the Fibonacci sequence $\{\Phi_{n,k}^{(s)}\}_{n=1}^\infty$. These numbers are defined by the relations: $$ \Phi_{n,k}^{(1)}=\sum_{j=0}^{n-1}\Phi_{j+1,k}\Phi_{n-j,k\,},\qquad \Phi_{n,k}^{(s)}=\sum_{j=0}^{n-1}\Phi_{j+1,k}\Phi_{n-j,k}^{(s-1)}\,,\quad s=2,3,...$$Moreover, $\Phi_{n,1}=F_n, \Phi_{n,1}^{(s)}=F_n^{(s)}$. An explicit formula for the representation of convolutions of k-sections of the Fibonacci sequence and a Binet type formula is established:$$\Phi_{n,k}^{(s)}=5^{-s}(F_k)^{-2s-1}\sum_{j=0}^{s}(-1)^{(k-1)j}{n+2s\choose j}{n+s-1-j\choose n-1} F_{k(n+2s-2j)}.$$ Several consequences were also obtained for $F_n$ and $F_n^{(s)}$, based on the connection between the derivatives of Chebyshev polynomials of the second kind $U_n(z)$ and their derivatives, as well as the connection for convolutions of k-sections of the Fibonacci sequence with derivatives of Chebyshev polynomials of the second kind via Lucas numbers $L_k$. Note that the sequences $\{\Phi_{n,k}^{(s)}\}_{n=1}^\infty$ for $k=3,4,...$ and $s=1,2,..$ are not included in the OEIS encyclopedia.