Umbral Calculus

It is about switching the index and the exponent to get the same results in a much easier way.
It can be formalized with the linear operator \(\mathcal{L}\) that maps a polynomial to a specific value. One can use \(\mathcal{L}^{-1}\) to transform the problem into the one with polynomials—which is easier—and solve it, and then transform back to the original problem with \(\mathcal{L}\).

1. Examples

For two sequences \(a_{n}, b_{n}\), To show \[ b_{n}=\sum_{k=0}^{n}\binom{n}{k}a_{k} \implies a_{n}=\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}b_{k} \] take the index and exponentiate by it. It immediately shows \[ b^{n}=(a+1)^{n} \implies a^{n}=(b-1)^{n}. \]

From the generating function of Bernoulli number \(B_{n}\)
\begin{align} \frac{x}{e^{x}-1}=&\sum_{k=0}^{\infty} \frac{B_{k}x^{k}}{k!} \\ \stackrel{\mathcal{L}^{-1}}{\implies} &\sum_{k=0}^{\infty} \frac{B^{k}x^{k}}{k!} \\ = &e^{Bx} \end{align}
and a linear transformation \(\mathcal{L}:B_{n}\mapsto B^{n}\), one finds
\begin{align} e^{-Bx}=& \sum_{k=0}^{\infty} \frac{(-1)^{k}B^{k}x^{k}}{k!} \\ \stackrel{\mathcal{L}}{\implies}& \sum_{k=0}^{\infty} \frac{B_{k}(-x)^{k}}{k!} \\ =&\frac{-x}{e^{-x}-1} = \frac{x}{e^{x}-1}e^{x} \\ \stackrel{\mathcal{L}^{-1}}{\implies}& e^{Bx}e^{x}=e^{(B+1)x} \\[1em] \implies& -B = B+1 \\ \implies& (-1)^{n}B^{n}=(B+1)^{n}=\sum_{k=0}^{n} \binom{n}{k}B^{n} \\ \stackrel{\mathcal{L}}{\implies}& (-1)^{n}B_{n} = \sum_{k=0}^{n} \binom{n}{k}B_{n} \\ \end{align}

First described by John Blissard in 1862, and studied by Édouard Lucas and James Sylvester.
Eric Temple Bell devised Umbral calculus in 1930s, to explain the hand-wavy calculations, where he defined umbral addition and umbral multiplication on umbral variables: \[ (a\oplus b)_{n}\mathop{=}^{\mathrm{def}} \sum_{k=0}^{n}\binom{n}{k}a_{k}b_{n-k} \] \[ (a \odot b)_{n}\mathop{=}^{\mathrm{def}}a_{n}b_{n} \] \[ c_{n}\mathop{=}^{\mathrm{def}}c^{n} \] but it was limited.
In 1970s, Gian-Carlo Rota and Steven Roman showed that it was a linear algebra.