Riordan Matrix Representations of Euler’s Constant γ and Euler’s Number e

Edray Herber Goins,Asamoah Nkwanta

doi:10.1155/2016/8324150

Abstract

We show that the Euler-Mascheroni constant γ and Euler’s number e can both be represented as a product of a Riordan matrix and certain row and column vectors.

Highlights

It was shown by Kenter [1] that the Euler-Mascheroni constant γ = lim n→∞ n [( ∑ m=1 1 m ) − ln n] ⋅ (1)
We show that the Euler-Mascheroni constant γ and Euler’s number e can both be represented as a product of a Riordan matrix and certain row and column vectors
Our ultimate goal is to explain how Riordan matrices are connected to a permutation representation π : G → GL(V) of a certain group G acting on an infinitedimensional vector space V

Summary

Introduction

Can be represented as a product of an infinite-dimensional row vector, the inverse of a lower triangular matrix, and an infinite-dimensional column vector:. Kenter’s proof uses induction, definite integrals, convergence of power series, and Abel’s Theorem. Kenter’s result follows from the identity ∑∞ m=1 Lm/m = γ, which in turn follows from an identity involving a definite integral. As another consequence of our main result, we can show that Euler’s number e lim Can be represented as a product of an infinite-dimensional row vector, a lower triangular matrix, and an infinitedimensional column vector. We consider the matrices as representations π : G → GL(V) of a certain group G, namely, the Riordan group, acting on an infinite-dimensional vector space V, namely, the collection of those formal power series h(x) in C⟦x⟧, where h(0) = 0

Introduction to Riordan Matrices

Proof of Kenter’s Result and Generalizations

Disclosure