Picturesque convolution-like recurrences and partial sums' generation

2025/07/31 by Gasparavičius, Ignas, Grigutis, Andrius, Petkelis, Juozas
#11B37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.23619

Abstract

Let \pmb b=\b₀, b₁, …\ be the known sequence of numbers such that b₀≠0. In this work, we develop methods to find another sequence \pmb a=\a₀, a₁, …\ that is related to \pmb b as follows: a_n=a₀ b_n+m+a₁ b_n+m-1+…+a_n+m b₀, n∈ℕ∪\0\, m∈ℕ. We show the connection of lim_n→∞a_n with a₀, a₁, …, a_m-1 and provide varied examples of finding the sequence \pmb a when \pmb b is given. We demonstrate that the sequences \pmb a may exhibit pretty patterns in the plane or space. Also, we show that the properly chosen sequence \pmb b may define \pmb a as some famous sequences, such as the partial sums of the Riemann zeta function, etc.

Related

• Laurent Phenomenon Sequences
• Automated Discovery and Proof of Congruence Theorems for Partial Sums of Combinatorial Sequences
• Convolutive sequences, I: Through the lens of integer partition functions
• A Chaotic Cousin Of Conway's Recursive Sequence
• Sums of Like Powers and Some Dense Sets