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

Technical Summary: Analytic Summation of Series Involving Higher-Order Derivatives of Chebyshev Polynomials of the Second Kind

Problem Statement
This paper addresses the analytic summation of functional series where the terms consist of higher-order derivatives of Chebyshev polynomials of the second kind, denoted as $U_N(z)$. Specifically, the authors investigate series where the degree of the polynomial is intrinsically linked to the order of the derivative. The primary challenge lies in determining the rational functions to which these series converge and establishing their connections to special numerical sequences generated by linear recurrence relations. While the properties of Chebyshev polynomials themselves are well-documented, series involving their higher-order derivatives remain less investigated.

Methodology
The authors employ a combination of generating functions, combinatorial identities, and analytic continuation to derive closed-form solutions.

1. Generating Functions: The study utilizes the generating function for Chebyshev polynomials of the second kind, $g(z, t) = (1 - 2tz + z^2)^{-1}$, and its higher-order partial derivatives with respect to the parameter $t$. This allows for the representation of derivatives $U^{(s)}_N(z)$ in terms of power series.
2. Combinatorial Analysis: The paper derives explicit formulas for the derivatives $U^{(s)}_{N+s-1}(z)$ and relates them to specific combinatorial coefficients. Lemma 3.1 establishes a summation identity involving binomial coefficients that facilitates the transformation of the series into rational functions.
3. Analytic Continuation: To handle formally divergent series, the authors apply analytic continuation to the boundary of the domain of convergence ($|z|=1$). This technique allows for the assignment of finite values to series that do not converge in the classical sense, linking them to classical Euler formulas.
4. Connection to Recurrent Sequences: The methodology explicitly maps the derived derivatives to Fibonacci numbers ($F_n$), Lucas numbers ($L_n$), Pell numbers ($P_n$), their convolutions ($F^{(s)}_n, P^{(s)}_n$), and $k$-sections of the Fibonacci sequence ($\Phi_{n,k}$).

Key Contributions and Results
The paper yields several significant theoretical results:

• Summation Formulas for Convergent Series: The authors derive closed-form expressions for two primary series for $|z| < 1$ and $|z| > 1$:
  $$\sum_{\sigma=0}^{\infty} \frac{z^\sigma U^{(\sigma)}_{N+\sigma-1}(z)}{2^\sigma \sigma!} = (1-z)^{-\frac{N+1}{2}} U_{N-1}\left(\frac{z}{\sqrt{1-z}}\right)$$
  $$\sum_{\sigma=0}^{\infty} \frac{z^{-\sigma} U^{(\sigma)}_{N+\sigma-1}(z)}{2^\sigma \sigma!} = \left(\frac{z}{z-1}\right)^{\frac{N+1}{2}} U_{N-1}\left(z\sqrt{\frac{z}{z-1}}\right)$$
  These sums are expressed as rational functions involving Chebyshev polynomials with composite arguments.

• New Identities for Numerical Sequences: By substituting specific values of $z$ (such as $z = i/2$ or $z = 1/(2L_k)$) into the general summation formulas, the paper establishes new closed-form identities for:

  • Sums involving convolved Fibonacci numbers ($F^{(s)}_n$) and convolved Pell numbers ($P^{(s)}_n$).
  • Sums involving $k$-sections of the Fibonacci sequence ($\Phi_{n,k}$).
  • Specific examples are provided for small $N$, yielding complex values for sums of convolved sequences.

• Summation of Divergent Series: Through analytic continuation, the paper obtains sums for formally divergent series. Notably, it recovers classical Euler formulas for alternating power series. For instance, the paper demonstrates that:
  $$\sum_{\sigma=0}^{\infty} (-1)^\sigma \binom{N+2\sigma}{N-1} = 2^{-N/2} \sin\left(\frac{N\pi}{4}\right)$$
  This result is generalized to arbitrary complex numbers using the Gamma function.

• Connection to Gegenbauer Polynomials: The paper notes that the derivatives $U^{(s)}_{N+s-1}(z)$ are related to Gegenbauer polynomials $C^{(\alpha)}_n(z)$ via the relation $\frac{1}{2^s s!} U^{(s)}_{N+s-1}(x) = C^{(s+1)}_{N-1}(x)$. Consequently, the summation formulas are re-expressed in terms of Gegenbauer polynomials, filling a gap in classical handbooks regarding series summed over the upper index.

Significance and Applications
The authors position this work as a contribution to the search for analytic representations of sums involving orthogonal polynomials. The significance of the results is twofold:

1. Theoretical: The work establishes exact connections between the derivatives of Chebyshev polynomials and number-theoretic sequences (Fibonacci, Lucas, Pell), providing new tools for deriving combinatorial identities.
2. Applied: The authors suggest potential applications in approximation theory, numerical methods utilizing orthogonal polynomials, and cryptography. Specifically, the relationship between Chebyshev polynomials and cryptographic systems (referencing the identity $U_{M-1}(T_n(x))U_{n-1}(x) = U_{nm-1}(x)$) is highlighted as a potential area for future utility.

The paper concludes that derivatives of Chebyshev polynomials serve as an effective mechanism for representing convolved recurrent sequences and that analytic continuation provides a rigorous framework for evaluating sums of divergent series that align with classical results.