Analytic summation of series involving higher-order derivatives of Chebyshev polynomials of the second kind and their applications to convolved linear recurrent sequences

TL;DR

This paper derives closed-form formulas for series involving derivatives of Chebyshev polynomials of the second kind and applies these to obtain identities for Fibonacci, Lucas, and Pell numbers.

Contribution

It introduces new analytic summation techniques for series of derivatives of Chebyshev polynomials and connects these to linear recurrence sequences and classical number identities.

Findings

Derived rational functions for series involving Chebyshev derivatives.

Established new identities for Fibonacci, Lucas, and Pell numbers.

Obtained sums for divergent series via analytic continuation.

Abstract

This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to determine the rational functions to which these series converge. These functions are expressed in terms of Chebyshev polynomials evaluated at a specific argument. Connections are established between derivatives of Chebyshev polynomials of the second kind and special numerical sequences generated by linear recurrence relations. New closed-form formulas are obtained for the sums of the series at various values of the argument. As consequences, combinatorial identities are derived for the Fibonacci, Lucas, and Pell numbers, for sections of the Fibonacci sequence, and for their convolutions. By means of analytic continuation, sums of formally divergent series are obtained, which in special cases correspond to the classical Euler formulas.