Umbral Calculus Derivation of the Bernoulli numbers

$$“(B-1)^n = B^n”$$

\begin{align*}
(B-1)^2 &= B^2\\
B^2 – 2B^1 + 1 &= B^2 \\
-2B^1+1 & = 0\\
B^1 &= \frac{1}{2} \\
B_1 &= \frac12
\end{align*}

\begin{align*}
(B-1)^3 &= B^3\\
B^3 – 3B^2 + 3B^1 – 1 &= B^3 \\
B^3 – 3B^2 + 3B_1 – 1 &= B^3 \\
-3B^2 + 3(\frac{1}{2})-1& = 0\\
-3B_2 + \frac{1}{2} &= 0 \\
B_2 &= \frac{1}{6}
\end{align*}

Thanks to Laurens Gunnarsen for showing me this strange trick.

I’ve finally understood the principle which allows us to lower the index. The step where we move $B^i$ to be $B_i$ is quite simple. As Rota and Roman say, one method of expressing an infinite sequence of numbers is by a transform method. That is, to define a linear transform $B$ such that $$B x^n = B_n$$

So, the above “lowering of the index” is actually using the relation $(X-1)^n = X^n$, and applying $B$ to both sides of it. To get $B(X-1)^n = B(X^n)$. Let’s look for example at the “lowering step” of the first calculation:
\begin{align*}
X^1 &= \frac{1}{2} \\
B (X^1) &= B(\frac12) \\
B_1 &= \frac12 B(1) = \frac12
\end{align*}

2 thoughts on “Umbral Calculus Derivation of the Bernoulli numbers”

1. Are you interested in a simple process to use linear symbolic algebra to generate the Appell polynomial sequence? Actually all of the Sheffer sequences; with a little more effort. The reason I found these constructions was to help make sense of Roman’s “Umbral Calculus” and Pascal’s Matrix. The Analogy is pretty much exact; that is any of Roman’s results can be rendered in terms of simple matrix algebra. In this case substituting the Pascal “Creation Matrix” for t in f(t)*exp(x*t) directly yield the coefficient matrix corresponding to the expansion in t for p(x).
The reason I like the matrix approach is that I trust the matrix operations of symbolic algebra; and besides, I do understand matrix operations.
Ray