[Paper Review] A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences

Stefano Barbero, Umberto Cerruti+1 more|arXiv (Cornell University)|Dec 15, 2012

Advanced Mathematical Theories and Applications5 references4 citations

TL;DR

This paper generalizes the binomial interpolated operator and analyzes its action on linear recurrent sequences, showing how it transforms characteristic polynomial roots and forming a group under composition. It identifies fixed sequences under these operators and derives explicit formulas linking Catalan, Fibonacci, Lucas, and triangular numbers through the operator's action.

ABSTRACT

In this paper we study the action of a generalization of the Binomial interpolated operator on the set of linear recurrent sequences. We find how the zeros of characteristic polynomials are changed and we prove that a subset of these operators form a group, with respect to a well–defined composition law. Furthermore, we study a vast class of linear recurrent sequences fixed by these operators and many other interesting properties. Finally, we apply all the results to integer sequences, finding many relations and formulas involving Catalan numbers, Fibonacci numbers, Lucas numbers and triangular numbers. 1

Motivation & Objective

• To generalize the binomial interpolated operator and study its action on linear recurrent sequences.
• To determine how the roots of characteristic polynomials change under this generalized operator.
• To prove that a subset of these operators forms a group under a defined composition law.
• To identify classes of linear recurrent sequences invariant under the operator.
• To apply the results to integer sequences, uncovering new relations among Catalan, Fibonacci, Lucas, and triangular numbers.

Proposed method

• Introduces a generalized binomial interpolated operator acting on sequences via a parameterized linear combination of sequence terms.
• Analyzes the transformation of the characteristic polynomial roots under the operator using algebraic manipulation and recurrence structure.
• Establishes a composition law between operators and proves closure, associativity, identity, and invertibility, confirming group structure.
• Identifies fixed sequences by solving functional equations derived from the invariance condition under the operator.
• Applies the operator to known integer sequences, deriving explicit identities and recurrence relations.
• Uses generating functions and polynomial root analysis to derive closed-form expressions for sequence transformations.

Experimental results

Research questions

• RQ1 How does the generalized binomial interpolated operator alter the roots of the characteristic polynomial of a linear recurrent sequence?
• RQ2 What algebraic structure do the set of such operators form under composition?
• RQ3 Which linear recurrent sequences remain invariant under the action of these operators?
• RQ4 What explicit relations can be derived between well-known integer sequences like Catalan, Fibonacci, and Lucas numbers using this operator?
• RQ5 How can the operator be used to generate new identities and closed-form expressions for sequences?

Key findings

• The generalized operator transforms the roots of the characteristic polynomial in a predictable algebraic manner, altering them via a rational function transformation.
• A subset of the generalized operators forms a group under composition, satisfying all group axioms with a well-defined identity and inverse.
• There exists a nontrivial class of linear recurrent sequences that are fixed by the operator, characterized by specific initial conditions and recurrence parameters.
• Explicit formulas are derived connecting Catalan numbers, Fibonacci numbers, Lucas numbers, and triangular numbers through the operator’s action.
• The operator enables the derivation of new identities among integer sequences by exploiting its algebraic structure and invariance properties.
• The transformation preserves the linear recurrence structure, allowing systematic generation of new sequences from known ones.