Application and Property of Many Special Types Recurrence Relation Polynomials in Number Theory and their Special Representation

It develops a formula that explicitly expresses the general term of a linear recurrent sequence, allowing us to generalize J. McLaughlin’s original finding on powers of 2 matrices to the case of a square matrix of size m ≤ 2 matrix. The identities of Fibonacci and Stirling numbers, as well as a variety of combinatorial relations, are deduced. It uses two-variable Hermit polynomials and their operational laws to derive integral representations of Chebyshev polynomials. Most of the Chebyshev polynomial properties can be obtained using the Hermit polynomials Hn(x, y) definitions and formalism. They also show how to use these results to introduce valid generalizations of these polynomial groups and derive new identities and integral representations for them. For Chebyshev polynomials of the first and second kinds, its present new generating functions. A recurrence relation is an important mathematical concept. Recurrence relations are used in a variety of fields, including mathematics, economics, physics, and other sciences. It presents a significant finding on the convergence of recurrence relation sequences as a function of the recurrence relation coefficient.

The authors declare no conflict of interest.

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The datasets used in this study are openly available at [repository link] and the source code is available on GitHub at [GitHub link].

This work did not receive any external funding.