Abel-Type Transformations and Telescoping Structures in Reciprocal Series of Second-Order Linear Recurrences

Technical Summary: Abel-Type Transformations and Telescoping Structures in Reciprocal Series of Second-Order Linear Recurrences

Problem Statement
The paper addresses the evaluation and transformation of infinite reciprocal series involving products of terms from second-order linear recurrence sequences (such as Fibonacci, Lucas, and general Horadam sequences). While numerous specific identities for such series exist in the literature—often referred to as "Brousseau sums" when involving products of consecutive terms—these results are typically derived using ad hoc methods tailored to specific sequences or parameter choices. The authors identify a lack of a general, systematic framework capable of unifying these disparate identities and extending them to series with three or more factors in the denominator.

Methodology
The core of the proposed approach is a discrete Abel-type summation formula (summation by parts). The authors utilize two primary identities from Lemma 1 to transform reciprocal sums into expressions exhibiting partial telescoping or alternating cancellation structures.

1. Transformation Mechanism: By setting $x_k$ and $y_k$ appropriately within the summation by parts formula, the authors convert a series of the form $\sum \frac{1}{w_{rk+l}w_{mk+s}w_{m(k+1)+s}}$ (where $w_n$ is a second-order recurrence) into an alternating sum.
2. Key Identity: The method relies heavily on the identity $w_t w_{q+1} - w_{t+1} w_q = (-1)^q E u_{t-q}$, where $u_n$ is the companion sequence and $E$ is a constant determined by initial conditions. This identity allows the inner sums to telescope.
3. Parity Dependence: A critical feature of the methodology is the distinction between even and odd values of the parameter $m$. The transformation formulas differ structurally depending on the parity of $m$, leading to qualitatively different representations (e.g., alternating signs appearing in the numerator for odd $m$ versus even $m$ in different contexts).
4. Extension to Higher Orders: The method is extended iteratively to handle series with four or more factors in the denominator, resulting in higher-order telescoping structures involving more consecutive terms of the underlying sequence.

Key Contributions and Results

• General Transformation Formulas: The paper establishes general theorems (Theorems 1, 4, 7, and 9) that provide transformation rules for infinite series of the form:
  $$\sum_{k=1}^{\infty} \frac{(\pm 1)^k}{w_{rk+l} w_{mk+s} w_{m(k+1)+s}}$$
  and their extensions to products of four or more terms. These formulas convert the original reciprocal series into alternating sums with built-in telescoping properties.
• Unification of Fibonacci and Lucas Identities: By specializing the general parameters to Fibonacci ($F_n$) and Lucas ($L_n$) numbers, the authors derive a wide array of known and new identities. For instance, they provide closed-form transformations for series like $\sum \frac{1}{F_k F_{k+1} F_{k+2}}$ and $\sum \frac{1}{L_k L_{k+1} L_{k+2}}$, expressing them as alternating sums of simpler terms.
• Applications to Combinatorial Sequences: The framework is applied to derive identities involving classical combinatorial sequences. Specifically, the authors present results for:
  • Harmonic Numbers ($H_n$): Transforming series involving $H_k$ in the numerator.
  • Catalan Numbers ($C_n$): Deriving identities involving binomial coefficients and Catalan numbers.
  • Stirling Numbers: Establishing relations for both Stirling numbers of the first and second kinds.
  • Chebyshev Polynomials: Providing identities for $T_n(x)$ and $U_n(x)$.
• Symmetric Identity: The authors demonstrate a symmetric property (Proposition 9) where the roles of the sequences in the summation by parts can be interchanged, yielding new dual identities for reciprocal series.

Significance and Claims
The paper claims to provide a "unified method" that systematically explains and produces identities for reciprocal series that were previously obtained through isolated, sequence-specific arguments. The significance of the work lies in:

1. Systematic Reduction: The ability to reduce complex multi-factor reciprocal sums to simpler forms with inherent telescoping or alternating structures.
2. Generality: The method applies uniformly to a broad class of second-order recurrence sequences (Horadam sequences), rather than being limited to Fibonacci or Lucas numbers.
3. Structural Insight: The approach naturally distinguishes between even and odd parameters, revealing structural properties of these series that were not previously highlighted in a unified manner.
4. Flexibility: The method serves as a flexible tool for generating new identities involving various combinatorial sequences, connecting reciprocal series with areas such as harmonic numbers and Stirling numbers.

The authors conclude that while future work could explore $q$-analogues or more general recurrence relations, the current framework successfully establishes a robust foundation for transforming and evaluating a wide class of infinite reciprocal series.