The Hirzebruch criterion for the Todd class

The Hirzebruch criterion for the Todd class is given in “The signature theorem: reminiscences and recreations” by Hirzebruch. The formal power series s(t) that defines the Todd class must satisfy \frac{d^n}{dt^n} (s(t))^{n+1}=n! . 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.

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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 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.]

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Bernoulli Appells

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

f(B.(x+1))-f(B.(x))={f}'(x)=D_x \; f(x) \; \; ,

where D_x= \frac{d}{dx} \; is the derivative w.r.t. x \; . 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.

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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:

\frac{\frac{d}{d(xD)}}{\frac{d}{d(:xD:)}}=\frac{\frac{d}{d(xD)}}{e^{\frac{d}{d(xD)}}-1}=\frac{\ln(1+\frac{d}{d(:xD:)})}{\frac{d}{d(:xD:)}}=\frac{nad_{\ln(D)}}{e^{nad_{\ln(D)}}-1}

=e^{B.(0)\frac{d}{d(xD)}}=e^{B.(0)\;nad_{\ln(D)}}

=\frac{1}{<\; e^{\bar{B}.(0)\frac{d}{d(xD)}}\;>}=\frac{1}{< \; e^{\bar{B}.(0)\;nad_{\ln(D)}}\; >}=\; <\;\frac{1}{1+\bar{B}.{(0)}\frac{d}{d(:xD:)}}\;>.

Decoding:

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Compositional Inversion and Appell Sequences

Compositional Inversion and Appell Sequences is a brief set of notes (pdf) on the relation between a Lagrange compositional inversion of a function in terms of the coefficients of the reciprocal of the function and a general set of Appell polynomials (special Scheffer sequences). The partition polynomials for the inversion form an Appell sequence w.r.t. the coefficient of the linear degree of the reciprocal and have coefficients related to a refinement of the Narayana numbers (A134264 / A001263) and  enumeration of Dyck (A125181) and Lukasiewicz paths, non-crossing partitions, and certain types of trees.

For a general discussion of non-crossing partitions and their intersection with other diverse domains of math, see “Generalized noncrossing partitions and combinatorics of Coxeter groups” by Drew Armstrong with extensive references.

For visual presentations of some combinatorics underlying many of the integer arrays here, see http://www.robertdickau.com/default.html#math.

At Play in an Orchard: Catalan, Riordan Appels and Lagrange Pairs reiterates the above notes and extends them to show how the Catalan numbers (OEIS-A000108)  are related to other special number sequences, such as the Fibonacci (A000045), Motzkin sums / Riordan (A005043), and Fine (A000957), through simple compositions, and through simple interpolation to associated polynomials, such as the Fibonacci polynomials  (A030528)  and those related to the Dyck / Lukasiewicz lattice paths, non-crossing partitions, and trees of A091867. In addition, it provides a procedure for constructing or identifying families related to other parent sequences.

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Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra

Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra: A brief note (pdf) on some relations among these topics, including the Catalan and Fuss-Catalan numbers. References are provided linking the analysis with the distribution of eigenvalues of random matrices, the Wigner semicircle law, and moduli spaces for marked Riemann surfaces.

For some notes on the Burgers equation (generalized form),  see Lectures on Partial Differential Equations (Chapter 4) by R. Salmon.

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Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform

Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform:   (pdf, under construction. ) Relations between differential operators represented in the two basis sets x^k (\frac{d}{dx})^k and x^k (\frac{d}{dx})_{x=0}^k and the underlying binomial transforms of the associated coefficients of the pairs of series are sketched as well as the relations among the Newton-Gauss interpolation of these coefficients, the action of the differential operators, and an associated modified Mellin and inverse Mellin transform pair. The associated Laguerre polynomials  and the Bell / Touchard  / Exponential polynomials are examined in this light. Connections of the associated Laguerre polynomials to the Witt Lie algebra and modular forms are established.

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