Tag Archives: Bernoulli polynomials

Compositional Inverse Operators and Sheffer Sequences

When considering operator inverses, one usually considers multiplicative inverses. As noted earlier in several entries, particularly, “Bernoulli and Blissard meet Stirling … ” (BBS), we see compositional inverse pairs of operators playing an important role in making associations among important … Continue reading

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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 … Continue reading

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Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion

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 … Continue reading

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The Bernoulli polynomials and Hirzebruch’s generalized Todd class

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. [This is done through … Continue reading

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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: Decoding:

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