There’s some confusion concerning representations of integration in the reciprocal spaces of the Fourier and Laplace transforms in some entries of MathOverflow that arises from not distinguishing among integration operators with different limits of integration (combined with some handwaving about group characters). Misassociation, or conflation, of distinct integration ops has been a source of historical confusion (cf. [Threefold Interpretation of Fractional Derivatives] by R. Hilfer as well as the comments in [MO question]). The convolution theorems provide a way to view differing integrations to effectively translate them into relatively simple factors in the reciprocal spaces.
The Riemann and Hurwitz zeta functions and the Mellin transform interpolation of the Bernoulli polynomials
This entry (expanding on the Bernoulli Appells entry) illustrates interpolation with the Mellin transform of the Bernoulli polynomials and their umbral inverses, the reciprocal polynomials, giving essentially the Hurwitz zeta function and the finite difference of , both of which can be umbrally inverted by the polynomials. It also elaborates on a set of generalized Bernoulli polynomials based on umbral composition of the Bernoulli polynomials with themselves and derives an “asymptotic” expression, or divergent series, for the Riemann zeta function noted in the Bernoulli Appells entry, which may be truncated to give very good approximations of the Hurwitz and Riemann zeta functions over ranges of parameters.
The raising op for any Appell sequence is determined by the derivative of the log of the e.g.f. of the basic number sequence, connecting the op to the combinatorics of the cumulant expansion OEIS-127671 of the moment generating function and its inverse relation A036040 for the general Bell polynomials of the Faa di Bruno formula for composition of functions. Diagrammatics of the partitions for the combinatorics of these entries can be found in the statistical physics references of A036040, but are by no means unique.
Furthermore, the combinatorics of the classical cumulants are, at a combinatorial level, intimately allied to that of the free cumulants of free probability theory and, consequently, to noncrossing partitions, as discussed by Jonathan Novak in “Three lectures on free probability”, Roland Speicher in “Free probability theory and non-crossing partitions”, Franz Lehner and coauthors in papers noted in the Bernoulli Appells entry, and by Ardila, Rincon, and Williams in “Positroids and noncrossing partitions”.
The Hirzebruch criterion for the Todd class is given in “The signature theorem: reminiscences and recreations” by Hirzebruch. The formal power series that defines the Todd class must satisfy . The e.g.f. for the Bernoulli numbers uniquely satisfies this criterion. I’d like to make a note of how the Bernoullis and the integer reciprocals are really two sides of the same coin and how both play a role in the Todd class criterion, and then note the relation to some important combinatorics, through a Lagrange inversion formula (LIF).
Let’s connect the Bernoullis, using their basic operational definition rather than their e.g.f., to the Todd genus and more through formal group laws (FGL, see note at bottom) and associated Lie ops and, therefore, compositional inversion again. [This is done through associating a power series to some basic FGLs, a series which Hirzebruch associates to genera–one of the main results of Hirzebruch’s book on Topological Methods of Alg. Geom., as he explicitly states on page 12 of his 2007 paper “Eulerian Polynomials”. The series is also naturally related to quantum groups explored by Hodges and Sukumar.]
The defining characteristic of the Bernoulli numbers operationally is that they are the basis of the unique Appell sequence, the Bernoulli polynomials, that “translate” simply under the generalized binomial transform (Appell property) and satisfy (for an analytic function, such as the exponential or logarithm, when convergent, or order by order for a formal power series) the umbral relation
where is the derivative w.r.t. . This then determines their umbral compositional inverse, the “reciprocal polynomials”, based on the reciprocal integers. From the derivative and translation property, the Euler-Maclaurin results follow easily as well as from the reciprocal e.g.f.s in operator form of the pair of Appell polynomials–that the e.g.f.s are reciprocals of each other, the operators are inverses, and the polynomials are umbral compositional inverses are inextricably linked. What isn’t evident from the Euler-Maclaurin perspective are the intimate associations to Lie theory, matrix reps, and simplices of this reciprocal pair of polynomials.
Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering
A set of identities that encapsulates relations among the Bernoulli numbers, the Stirling numbers of the first and second kinds, and operators related to the umbral calculus of Blissard and his contemporaries: