Pascal's Pyramid and the Fibonacci Operator Framework: Complex Extensions, Potential Wells, Spectral Classification, and Rigorous Error Bounds
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2025
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| _version_ | 1866902056253521920 |
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| author | Betzer, David |
| author_facet | Betzer, David |
| contents | <p dir="auto">This paper presents a unified, rigorously proven matrix-analytic framework linking Pascal’s pyramid to Fibonacci partial sums via the partial sum operator S(i).</p> <p dir="auto">We establish the core identity P = P_L P_U, where P_L and P_U are the lower and upper triangular Pascal matrices, reconstructing binomial coefficients exactly.</p> <p dir="auto">The global decomposition P_L X = A x + B with A + B = P_L 1 = 1 connects to the Golden Equation framework where α(k) + α(k)^{-k} = 2 for the family of k-bonacci growth constants.\n\n</p> <p dir="auto">We prove both S(i) = P_L^i (backward cumulative sums) and S(i) = P_U^i (forward cumulative sums) via generating functions and induction, with full finite-dimensional corrections. Continuous extensions P_k = e^{k ln P} for k ∈ C unify damped oscillatory behavior. The potential V(λ; k) = e^λ + e^{-kλ} forms stable wells at λ_0 = ln k / (k+1).\n\n</p> <p dir="auto">For amplified recurrences r_m = m r_{m-1} + ⋯ + 1, we derive the dominant root β_m with explicit, computable error bounds:\n|β_m - q(m)| ≤ 1/(m(m-1)), q(m) = 1/2 [m + 1 + √((m+1)^2 - 4)], m ≥ 2.\n\n</p> <p dir="auto">The eigenvalue map k_i = -ln(2 - λ_i)/ln λ_i enables O(n) spectral classification. All finite Pascal matrices P_n are invertible with det(P_n) = 1.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17528956 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Pascal's Pyramid and the Fibonacci Operator Framework: Complex Extensions, Potential Wells, Spectral Classification, and Rigorous Error Bounds Betzer, David Pascal matrix Fibonacci sequence Golden ratio LU decomposition Spectral classification number theory error bounds Generating functions Linear recurrences Algebra Operator algebra Linear algebra algebra Mathematics Mathematics Pure mathematics Applied mathematics mathematics Mathematical physics physics <p dir="auto">This paper presents a unified, rigorously proven matrix-analytic framework linking Pascal’s pyramid to Fibonacci partial sums via the partial sum operator S(i).</p> <p dir="auto">We establish the core identity P = P_L P_U, where P_L and P_U are the lower and upper triangular Pascal matrices, reconstructing binomial coefficients exactly.</p> <p dir="auto">The global decomposition P_L X = A x + B with A + B = P_L 1 = 1 connects to the Golden Equation framework where α(k) + α(k)^{-k} = 2 for the family of k-bonacci growth constants.\n\n</p> <p dir="auto">We prove both S(i) = P_L^i (backward cumulative sums) and S(i) = P_U^i (forward cumulative sums) via generating functions and induction, with full finite-dimensional corrections. Continuous extensions P_k = e^{k ln P} for k ∈ C unify damped oscillatory behavior. The potential V(λ; k) = e^λ + e^{-kλ} forms stable wells at λ_0 = ln k / (k+1).\n\n</p> <p dir="auto">For amplified recurrences r_m = m r_{m-1} + ⋯ + 1, we derive the dominant root β_m with explicit, computable error bounds:\n|β_m - q(m)| ≤ 1/(m(m-1)), q(m) = 1/2 [m + 1 + √((m+1)^2 - 4)], m ≥ 2.\n\n</p> <p dir="auto">The eigenvalue map k_i = -ln(2 - λ_i)/ln λ_i enables O(n) spectral classification. All finite Pascal matrices P_n are invertible with det(P_n) = 1.</p> |
| title | Pascal's Pyramid and the Fibonacci Operator Framework: Complex Extensions, Potential Wells, Spectral Classification, and Rigorous Error Bounds |
| topic | Pascal matrix Fibonacci sequence Golden ratio LU decomposition Spectral classification number theory error bounds Generating functions Linear recurrences Algebra Operator algebra Linear algebra algebra Mathematics Mathematics Pure mathematics Applied mathematics mathematics Mathematical physics physics |
| url | https://doi.org/10.5281/zenodo.17528956 |