Rings and Modules Seminar: March 19, 2009

Abstract:

Define B⁰ := 1 and (B+1)ⁿ – Bⁿ = 0 for n ≥ 2. That is,

B₀ = 1, B₁ = –1/2, B₂ = 1/6, B₃ = 0, B₄ = 1/30, B₅ = 0, …. We refer to B_k as the k^th Bernoulli number. Let us rewrite this definition into one equation: e^(B+1)x – e^Bx = x. Hence, x = e^(B+1)x – e^Bx = e^Bxe^x – e^Bx1 = e^Bx(e^x – 1), and so, e^Bx = x/(e^x – 1). But x/(e^x – 1) + x/2 is an even function, and so B_2n+1 = 0 for n ≥ 1.

This is the basic Umbral Calculus argument for the Bernoulli numbers; I will explain Umbral Calculus in terms of modern algebra.

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Department of Mathematics
Server

Rings and Modules Seminar
~ Abstracts ~

Department of Mathematics Server

Rings and Modules Seminar ~ Abstracts ~

Department of Mathematics
Server

Rings and Modules Seminar
~ Abstracts ~