Convolved numbers of k-section of the Fibonacci sequence: properties, consequences

Abstract

Chebyshev polynomials are used to solve numerous applied problems in computer science involving interpolation theory, approximation theory, numerical analysis, dynamical systems theory, and number theory. However, when forming pseudo-random recurrent sequences, the use of derivatives of Chebyshev polynomials, especially of higher orders, is much less common in the literature, although the issue is quite actual. This article somewhat fills this gap. It is shown that among recurrent sequences used for information analysis and to improve its cryptographic protection, the Fibonacci sequence and its generalizations are the most popular. This article considers a further generalization of Fibonacci numbers, namely, folded numbers k-sections of the Fibonacci sequence. The objective of the research is to further generalize the Fibonacci numbers, namely, the collapsed numbers of the k-intersection of the Fibonacci sequence. The research used modern methods of number theory. The properties of the obtained sequences are determined, and new connections between their elements are found. A further development of Fibonacci-type sequences is proposed, based on the relationship between the derivatives of Chebyshev polynomials of the second kind and Chebyshev polynomials themselves, as well as on the relationship between the folded numbers  the cross-sections of k the Fibonacci sequence and the derivatives of Chebyshev polynomials of the second kind through Lucas numbers. A number of identities are obtained linking Fibonacci numbers and Lucas numbers. It is shown that higher-order derivatives of Chebyshev polynomials prove to be an effective basis for solving certain problems in number theory, namely, the construction of new sequences. Thus, the study resulted in a family of generalized sequences with higher order growth and more complex coupling coefficients than the known Fibonacci sequences. This makes the resulting generalized sequences ideal for sparse data compression and for solving a number of problems in information security. It is proven that the resulting sequences are original and are not presented in the OEIS encyclopedia, confirming the potential of the proposed approach to the formation of various sequences that can be used to improve the reliability of information systems.