Bernoulli numbers, Faulhaber’s formula, 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

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.

    The Algebraic Justification: A 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)

    000000$. 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.

    A Deeper Realization: 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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