Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators

TL;DR

This paper characterizes fixed sequences for the Generalized Binomial operator and related operators using functional equations, providing applications to well-known integer sequences and their identities.

Contribution

It offers a complete characterization of fixed sequences for the Generalized Binomial operator and explores their relations with other operators and classical sequences.

Findings

Fixed sequences for the Generalized Binomial operator are characterized.

Connections between fixed sequences and classical sequences like Fibonacci and Catalan are established.

Eigen-sequences for compositions of operators are identified.

Abstract

This paper is devoted to the study of eigen-sequences for some important operators acting on sequences. Using functional equations involving generating functions, we completely solve the problem of characterizing the fixed sequences for the Generalized Binomial operator. We give some applications to integer sequences. In particular we show how we can generate fixed sequences for Generalized Binomial and their relation with the Worpitzky transform. We illustrate this fact with some interesting examples and identities, related to Fibonacci, Catalan, Motzkin and Euler numbers. Finally we find the eigen-sequences for the mutual compositions of the operators Interpolated Invert, Generalized Binomial and Revert.