Bernoulli numbers, Umbral Calculus, Volkenborn integrals

Within the study of special functions and number theory, certain notational conventions occasionally arise that are so elegant and effective they appear to be a kind of magic. One of the most beautiful examples of this is the umbral calculus developed for Bernoulli numbers, where the indexed numbers B_k are formally treated as powers B^k of a single entity, B. This seemingly arbitrary substitution transforms a wide array of complex identities into familiar results from elementary algebra, providing a powerful mnemonic and a window into a deeper structure.

The Umbral Correspondence

B_k \longrightarrow B^k

The power of this “formal variable” approach is best seen through examples where intricate formulas are mapped to conspicuously simple algebraic statements.

Faulhaber’s Formula for Sums of Powers: The formula for the sum of the first n p-th powers is a complicated polynomial in n. The following expansion defines Bernoulli numbers in terms of the polynomials for sums of powers of integers:

\displaystyle 1^p+2^p+\cdots +n^p=\frac{1}{p+1} \sum_{j=0}^p\left(\begin{array}{c}p+1 \\ j \end{array}\right) B_j n^{p+1-j}

If we formally think of B_j as B^j this formula becomes

\displaystyle 1^p+2^p+\cdots +n^p=\frac{1}{p+1} \sum_{j=0}^p\left(\begin{array}{c}p+1 \\ j \end{array}\right) B_j n^{p+1-j}= \frac{1}{p+1} \sum_{j=0}^p\left(\begin{array}{c}p+1 \\j \end{array}\right) B^j n^{p+1-j} =

\displaystyle =\frac{(B+n)^{p+1}-B^{p+1}}{p+1} = \int_{B}^{B+n} x^p dx

This allows us to write the sum of powers as a formal integral: \displaystyle \int_{B}^{B+n} x^p dx

The formal replacement of B_j with B^j has many nice properties. For instance, the generating series of Bernoulli numbers

\displaystyle \frac{x}{e^x-1} =\sum_{k=0}^{\infty}\frac{B_k x^k}{k!}

can be thought of as

\displaystyle \frac{x}{e^x-1} =\sum_{k=0}^{\infty}\frac{B_k x^k}{k!} =e^{Bx}

The Bernoulli polynomial \displaystyle B_n(x)=\sum_{k=0}^{n} {n \choose k} B_{n-k} x^k can seen as

\displaystyle B_n(x)=\sum_{k=0}^{n} {n \choose k} B_{n-k} x^k =(B+x)^n

Consider the formula:

\displaystyle B_n(y+x)=\sum_{k=0}^n\left(\begin{array}{l} n \\k \end{array}\right) B_{n-k}(y) x^k

In terms of formal expressions, this equation will be mapped to

\displaystyle (B+y+x)^n = \sum_{k=0}^n\left(\begin{array}{l} n \\k \end{array}\right) (B+y)^{n-k} x^k

which is obvious (Binomial expansion). So proving the original formula is trivial once we notice this formal similarity.

Similarly the formula \displaystyle \frac{d}{d x} B_n(x)=n B_{n-1}(x) follows from the equality

\displaystyle \frac{d}{d x} (B+x)^n =n (B+x)^{n-1}

The justification for this comes from the fact that we use the map B^k \to B_k everywhere and do some linear manipulations. We can prove many identities for the Bernoulli numbers and polynomials using this map from polynomials in the formal variables B to Bernoulli numbers.

Linear Functional: This notational sleight-of-hand can be placed on a rigorous footing by considering the map as a linear functional. Let us define a linear operator, \mathbf{L}, that acts on polynomials in a formal variable B and maps them to numbers by the rule \mathbf{L}(B^k) = B_k. For example, \mathbf{L}((B+x)^n) = B_n(x).

All the identities above work because they involve operations (like differentiation with respect to x, or binomial expansion) that are linear and commute with the operator \mathbf{L}. For instance, the derivative property holds because:

\displaystyle \frac{d}{dx} B_n(x) = \frac{d}{dx} \mathbf{L}((B+x)^n) = \mathbf{L}\left(\frac{d}{dx}(B+x)^n\right) = \mathbf{L}(n(B+x)^{n-1}) = nB_{n-1}(x).

The “formal similarity” is, in reality, an isomorphism between algebraic structures, and the umbral notation is a highly efficient way to manage this isomorphism.

The Volkenborn Integral

While the linear functional approach provides an algebraic justification, a deeper perspective comes from the world of p-adic analysis. The abstract operator \mathbf{L} can be given a concrete analytic meaning as an integral over the p-adic integers, \mathbb{Z}_p. This is the Volkenborn integral.

For a continuous function f on \mathbb{Z}_p, its Volkenborn integral is defined as a limit of Riemann sums. A remarkable fact is that for the monomial x^k, this integral gives precisely the k-th Bernoulli number: \displaystyle \int_{\mathbb{Z}_p} x^k \mathrm{~d} x=B_k

So a way to think of this linear map is the Volkenborn integral:

\displaystyle \int_{\mathbb{Z}_p} B^k \mathrm{~d} B=B_k

Just think of integrating the B variables in this Volkenborn sense to get Bernoulli numbers. From this viewpoint, the umbral operator \mathbf{L} is realized as integration on \mathbb{Z}_p (normalized to have total volume 1). The formal variable B can be thought of as the integration variable in this context. The umbral method, which at first appears to be a mere notational convenience, is revealed to be a shadow of a deep arithmetic structure, connecting the combinatorial properties of Bernoulli numbers to the analytic and topological properties of the p-adic integers.

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