Bernoulli, Ramanujan, Toeplitz and the triangular matrices

TL;DR

This paper explores the connection between Bernoulli numbers and lower triangular Toeplitz systems, introduces efficient solution methods for these systems, and relates Ramanujan's sparse systems to Toeplitz matrices.

Contribution

It establishes a link between Bernoulli numbers and Toeplitz systems and develops efficient algorithms for solving such systems when their size is a power of 2 or 3.

Findings

Bernoulli numbers solve specific Toeplitz systems

Ramanujan's sparse systems are equivalent to Toeplitz systems

Efficient solution methods are provided for systems with sizes as powers of 2 or 3

Abstract

By using one of the definitions of the Bernoulli numbers, we prove that they solve particular odd and even lower triangular Toeplitz (l.t.T.) systems of equations. In a paper Ramanujan writes down a sparse lower triangular system solved by Bernoulli numbers; we observe that such system is equivalent to a sparse l.t.T. system. The attempt to obtain the sparse l.t.T. Ramanujan system from the l.t.T. odd and even systems, has led us to study efficient methods for solving generic l.t.T. systems. Such methods are here explained in detail in case n, the number of equations, is a power of b, b=2,3 and b generic.