Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1508.07894

On the Product Representation of Number Sequences, with Application to the Fibonacci Family

Michelle Rudolph-Lilith
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate general properties of number sequences which allow explicit representation in terms of products. We find that such sequences form whole families of number sequences sharing similar recursive identities. Restricting to the cosine of fractional angles, we then study the special case of the family of $k$-generalized Fibonacci numbers, and present general recursions and identities which link these sequences.

Explore related subjects

math.NTmath.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michelle Rudolph-Lilith. 2015-08-25. On the Product Representation of Number Sequences, with Application to the Fibonacci Family. https://arxiv.org/abs/1508.07894

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Documentation for the ratpoints program

This note explains how to obtain, install, and use the ratpoints program. The program finds rational points up to a specified height on hyperelliptic curves using a highly optimized quadratic sieving algorithm.

math.NT↗
arXiv2026

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove a two-term asymptotic formula for the eigenvalues of L, show their algebraic simplicity, sign alternation pattern, and decrease in absolute value. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.

math.NT↗
arXiv2026

Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Let $\mathbb{WP}^n_{\mathbf{q}}$ be a weighted projective space with weights $\mathbf{q} = (q_0, \dots, q_n)$, $q = \operatorname{lcm}(q_i)$, and let $ϕ\colon \mathbb{WP}^n_{\mathbf{q}} \to \mathbb{P}^n$, $[x_i] \mapsto [x_i^{q/q_i}]$, be the Veronese morphism. A point of $\mathbb{P}^n(\mathbb{Q})$ is the image of a rational point of $\mathbb{WP}^n_{\mathbf{q}}$ only if its valuation vector at every prime satisfies a Kummer congruence. We count the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ of bounded weighted height $\mathfrak{h} = H(ϕ(\,\cdot\,))^{1/q}$ and prove that, on the locus where all coordinates are nonzero, $$ Z^{\circ}_{\mathfrak{h}}\big( \mathbb{WP}^n_{\mathbf{q}}(\mathbb{Q}), X \big) = X^{q\,a(\mathbf{q})} P_{\mathbf{q}}(\log X) + O\big( X^{q\,a(\mathbf{q}) - θ} \big), \qquad θ> 0, $$ with $P_{\mathbf{q}}$ of exact degree $β(\mathbf{q})$, where $a(\mathbf{q})$ and $β(\mathbf{q})$ are the value and the dimension of the optimal face of a linear program determined by the Kummer congruences. The exponent satisfies $Q \leq q\,a(\mathbf{q}) \leq q(n+1)$, $Q = \sum q_i$, with equality on the right if and only if the exponents $q/q_i$ are pairwise coprime; the difference $q(n+1) - q\,a(\mathbf{q})$ measures the sparsity of the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ relative to those of its Veronese image. The leading constant is evaluated when the dual optimum is diagonal and for the weights $(2,2,3,3)$. The full counting function follows by stratification, and we formulate the conjecture over an arbitrary number field.

math.NT↗