# Collected Solutions to Open Problems in Fibonacci Sequences **论文集标题**:Collected Solutions to Open Problems in Fibonacci Sequences --- ## 2026i ### Title Self-Contained Matrix–Eigenvalue Solutions for Three Basic Fibonacci Quarterly Problems B-1361–B-1363 ### Abstract We present self-contained solutions to three basic problems from The Fibonacci Quarterly using a unified companion-matrix framework. The first problem concerns centroid identities in triangles with coefficients from Fibonacci and Lucas sequences. We prove that a weighted side-length relation is equivalent to a weighted centroid-distance relation, and that the isosceles condition reduces to a simple equality. The second problem is an arctangent infinite series involving odd Lucas numbers; we establish the exact splitting that reduces the series to a lacunary identity. The third problem is a logarithmic inequality involving Fibonacci and Lucas numbers; we reduce it to the elementary arithmetic-geometric mean estimate. All proofs proceed from first principles and use no external theorems. The underlying mechanism is then extended to its natural general form: arbitrary recurrence orbits for the centroid identity, positive orbits of a cubic map for the arctangent series, and finite-dimensional Cauchy inequalities for the logarithmic problem. ### Keywords Fibonacci number; Lucas number; companion matrix; eigenvalue method; triangle centroid; arctangent series; logarithmic inequality; self-contained proof; tripling map; logarithmic Cauchy inequality --- ## 2026j ### Title Self-Contained Matrix–Eigenvalue Solutions for Two Advanced Fibonacci Quarterly Problems H-956 and H-957 ### Abstract We give complete solutions to two advanced problems from The Fibonacci Quarterly using the companion-matrix method. For the first problem, which involves exponential generating functions of Fibonacci and Lucas numbers, we evaluate the tail-weighted sums in closed form. The solution uses Binet formulas, adjacent product identities, and the trace of a matrix exponential. For the second problem, we solve the explicit representation of sequences satisfying a general second-order recurrence with variable coefficients. By normalizing the recurrence, we reduce it to standard Fibonacci and Lucas polynomial forms, and obtain the corresponding Viète and Viète-Lucas polynomial representations as special cases. We then extend both results to their full generality: the tail-weighted sum is expressed as a Kronecker product of linear systems, and the second-order recurrence is handled over arbitrary commutative rings. The terminal problem of spectral compression is completely characterized by Jordan projection coefficients. ### Keywords Fibonacci number; Lucas number; exponential generating function; companion matrix; matrix exponential; Kronecker product; spectral projection; Jordan decomposition; spectral compression; Fibonacci polynomial; Lucas polynomial; Viète polynomial; second-order recurrence; universal recurrence --- ## 2026k ### Title Self-Contained Matrix Solutions for Three Elementary Fibonacci Quarterly Problems B-1366–B-1368 ### Abstract We solve three elementary problems from The Fibonacci Quarterly using a unified matrix method. The first problem evaluates two arctangent infinite series involving odd Fibonacci numbers. Using an exact arctangent splitting derived from Fibonacci identities, both series collapse to boundary terms. The second problem establishes a binomial convolution identity for any Fibonacci-type sequence; the proof uses finite telescoping along a companion orbit and requires no generating functions. The third problem proves a logarithmic inequality with Fibonacci and Lucas numbers; the product simplifies exactly to a known Fibonacci term plus a positive correction. Each solution is extended to its natural terminal form: the arctangent cancellation becomes a complete shift-filter classification, the binomial convolution becomes a universal identity in a quotient ring that characterizes the Fibonacci recurrence, and the logarithmic inequality becomes a two-base mixing formula with explicit equality conditions. ### Keywords Fibonacci number; Lucas number; companion matrix; eigenvalue method; arctangent series; shift filter; binomial convolution; Fibonacci-type sequence; universal recurrence; logarithmic inequality; logarithmic mixing; telescoping sum --- ## Collected Volume Information **Title**: Collected Solutions to Open Problems in Fibonacci Sequences **Author**: Jianming Wang **Description**: This volume collects three self-contained papers that solve eight open problems from The Fibonacci Quarterly using a unified companion-matrix framework. The problems span basic and advanced levels, covering arctangent series, logarithmic inequalities, centroid geometry, binomial convolutions, exponential generating functions, and second-order recurrence representations. Each solution is accompanied by a terminal extension that lifts the concrete mechanism to its natural general form, including shift-filter classification, quotient-ring characterization, and spectral compression. All proofs are independent and proceed from first principles.
Jianming Wang (Mon,) studied this question.