A Method for Uniformly Proving a Family of Identities

TL;DR

This paper introduces a proof method for uniformly establishing identities across families of recursive sequences, demonstrated through a specific sequence family and a recursive triangle structure.

Contribution

The paper develops a novel proof technique for families of identities in recursive sequences, utilizing characteristic polynomial divisibility, with a case study on a parameterized sequence family.

Findings

Established a general identity for the family of recursive sequences.

Defined a recursive triangle satisfying a specific recursion.

Proved the divisibility of characteristic polynomials in the identities.

Abstract

This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the family of recursive sequences $F^{(k)}_n = \sum_{i=1}^k F^{(k)}_{n-i}, n \ge 0, k \ge 2,$ with $n$ a parameter varying over integers, and $k$ a parameter indexing members of the family. The main theorem states $ F^{(k)}_n = \sum_{j=1}^k P_{k,j} F^{(k)}_{n-jk},$ with $P$ a recursive triangle satisfying the triangle recursion $P_{i,j}=2P_{i-1,j}- P_{i-1,j-1},$ with appropriate initial conditions. The proof of the theorem exploits the fact that characteristic polynomials of identities are divisible by the characteristic polynomial of the recursion generating the underlying sequence.