2026 Series n–q: English Titles, Abstracts, and Keywords

# 2026 Series n–q: English Titles, Abstracts, and Keywords ## 2026n**Title:** Self-Contained Matrix–Eigenvalue Solutions for Fibonacci Quarterly Advanced Problems H-941–H-943 **Abstract:** This paper solves three advanced problems from Fibonacci Quarterly using a unified companion matrix framework. H-941 is a finite tensor-observable coboundary verified by six spectral values; H-942 is an asymptotic integral lower bound obtained from Stirling's formula and trigonometric convexity; H-943 is a Lucas logarithmic zeta function whose meromorphic continuation and residue are extracted from the decomposition log L₂ₙ=2n log alpha + log(1+alpha⁻⁴ⁿ). The universal extensions include finite-dimensional tensor coboundaries, regularly varying factorial-root endpoints, and twisted log-polynomial zeta continuations. **Keywords:** Fibonacci number; Lucas number; companion matrix; tensor observable; finite telescoping; Stirling formula; logarithmic zeta function; Hurwitz zeta function; analytic continuation --- ## 2026o**Title:** Self-Contained Matrix–Sequence Solutions and Universal Extensions of Fibonacci Quarterly Problems B-1349 and B-1351 **Abstract:** Two Fibonacci Quarterly problems are solved by elementary matrix methods. B-1349 reduces via F₂ₙ=F_nL_n and L_n=2Fₙ₊₁-F_n to Jensen convexity of x log x. B-1351 is an arctangent series for Pell-type sequences; the correct telescope is obtained from the eigenvalue boundary angles of the companion matrix. The universal extensions lift B-1349 to Markov-chain entropy dissipation and Bregman Jensen gaps, and B-1351 to higher-step and spectral arctangent coboundaries for commuting positive semidefinite pencils. **Keywords:** Fibonacci number; Lucas number; Pell-type recurrence; companion matrix; Jensen convexity; relative entropy; Markov kernel; Bregman divergence; arctangent series; spectral telescope --- ## 2026p**Title:** Self-Contained Matrix–Eigenvalue Solutions and Universal Extensions for Four Advanced Fibonacci Quarterly Problems H-936–H-939 **Abstract:** Four advanced problems from Fibonacci Quarterly are solved by companion-matrix methods. H-936 establishes a ceiling continued fraction for e-1/r via Engel-sum transfer; H-937 proves a geometric-mean inequality by two-layer convexity; H-938 gives a reciprocal-square tail certificate; H-939 yields finite Fibonacci–Lucas determinant splittings with parity-controlled telescoping. Universal extensions include seed-filter coboundaries, weighted Lucas determinant identities, finite Hankel-rank classification, infinite-memory companion operators, Banach resolvent summation, cyclic Fourier filter banks, and infinite-boundary Lucas splittings. **Keywords:** Fibonacci number; Lucas number; companion matrix; ceiling continued fraction; Engel expansion; geometric mean; Hankel rank; Banach resolvent; Fourier filter bank; parity telescoping --- ## 2026q**Title:** Self-Contained Matrix and Seed–Filter Generalizations of Three Fibonacci Quarterly Problems B-1346–B-1348 **Abstract:** Three elementary Fibonacci–Lucas problems are solved and extended through matrix and seed-filter language. B-1346 is a reciprocal Lucas inequality sharpened to all real unit Lucas orbits. B-1347 is a binomial identity generalized to every Fibonacci-recurrence sequence with an exact boundary correction. B-1348 is a divisibility modulo 7 statement recovered from the eigenvalue ratio criterion for finite fields. Universal extensions include infinite-dimensional Fibonacci operators, finite automaton recognition of periodic zero sets, and the exact finite-group cocycle period formula for discrete integrals. **Keywords:** Fibonacci number; Lucas number; companion matrix; seed-filter system; unit Lucas orbit; binomial identity; finite field; eigenvalue ratio; automatic set; Fibonacci operator; finite cocycle --- ## Collected Volume: Fibonacci Quarterly Open Problems Solved — 2026 Series n–q This volume collects four papers (2026n–2026q) in the 2026 Series, each solving multiple advanced problems from Fibonacci Quarterly using a unified companion-matrix and seed-filter framework. The series demonstrates that diverse problem types—finite telescoping, integral asymptotics, logarithmic zeta functions, Markov entropy dissipation, arctangent telescopes, continued fractions, geometric means, determinant splittings, binomial identities, and modular zero classifications—are all governed by the same finite-state orbit, block drift, and filter stabilization principles. The solutions are self-contained and systematically extended to universal operator, automata-theoretic, and functional-analytic settings.