## 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 $\frac{d^n}{dt^n} (s(t))^{n+1} |_{t=0} = n!$ . 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 (LIF).

At the heart of this is an Appell sequence pair for the Bernoulli polynomials and the normalized reverse face polynomials of the simplices (see the entry Bernoulli Appells for background) with their base number sequences being the Bernoullis $B_n$ and the reciprocals $\bar{B}_n = 1/(n+1)$ with e.g.f.s

$\displaystyle e^{B.t}=\frac{t}{e^t-1}, \;\;\;\; e^{\bar{B}. t}=\frac{e^t-1}{t}.$

Consider a function $f(x)$ with compositional inverse $f^{-1}(x)$ such that $f(0)=0$ and $f_x(0)=1$ (this is necessary for the associated Appell sequences to have nice properties). Let $z\;u=f^{-1}(\nu)$. Then $\nu=f(z\; u)$ and $z\; du = D_{\nu}f^{-1}(\nu)\;d\nu=D_{\nu}f^{-1}(\nu)\;z\; f^{'}(z\;u)\; du$, the inverse function theorem, and the formal Borel-Laplace transform under a change of variables (for suitable functions) is a weighting of the inverse function relation giving the standard LIF

$\displaystyle \int_{0}^{\infty }\frac{1}{z} \exp\left ( -\nu \frac{1}{z} \right )D_{\nu}f^{-1}(\nu)\;d\nu=\int_{0}^{\infty }\exp\left [ - \frac{f\left (z\;u \right ) }{z}\right ]\;du$

$\displaystyle =\int_{0}^{\infty }\exp[z\;u\;D_{\omega=0}]\;\exp\left [ -u \; \frac{f(\omega)}{\omega} \right ]\;du$

$\displaystyle =\sum_{n\geq 0}z^n\;D^n_{\omega=0}\int_{0}^{\infty }\frac{u^n}{n!}\;\exp\left [ -u \; \frac{f(\omega)}{\omega} \right ]\;du$

$\displaystyle =\sum_{n\geq 0}z^n\;D^n_{\omega=0}\left [ \frac{\omega}{f(\omega)} \right ]^{n+1},$

so we have the LIF with four key expressions, but here we focus on just three

$\displaystyle \int_{0}^{\infty }\frac{1}{z} \exp\left ( -\nu \frac{1}{z} \right )D_{\nu}f^{-1}(\nu)\;d\nu=\sum_{n\geq 0}z^n\;D^n_{\omega=0}\int_{0}^{\infty }\frac{u^n}{n!}\;\exp\left [ -u \; \frac{f(\omega)}{\omega} \right ]\;du$

$\displaystyle=\sum_{n\geq 0}z^n\;D^n_{\omega=0}\left [ \frac{\omega}{f(\omega)} \right ]^{n+1}.$

With $f(x)=e^x-1$ and $f^{-1}(x)=ln(1+x)$,

$\displaystyle \int_{0}^{\infty }\frac{1}{z} \exp\left ( -\nu \frac{1}{z} \right )\frac{1}{1+\nu}\;d\nu=\sum_{n\geq 0}(-1)^{n} n! \; z^n$

$\displaystyle =\sum_{n\geq 0}z^n\;D^n_{\omega=0}\int_{0}^{\infty }\frac{u^n}{n!}\;\exp\left [ -u \; \frac{e^{\omega}-1}{\omega} \right ]\;du$

$\displaystyle =\sum_{n\geq 0}z^n\;D^n_{\omega=0}\left [ \frac{\omega}{e^{\omega}-1} \right ]^{n+1}.$

Equivalently,

$\displaystyle (-1)^{n} =\; \frac{D^n_{\omega=0}}{n!}\int_{0}^{\infty }\frac{u^n}{n!}\;\exp\left [ -u \; e^{\bar{B}. \omega} \right ]\;du= \frac{D^n_{\omega=0}}{n!}\left [e^{B.\;\omega} \right ]^{n+1}.$

Now we can see how the reciprocal derivatives in the inverse fct. theorem $f_u(u)=\frac{1}{f^{-1}_{\nu}{(\nu)}}$ are related to the reciprocal Appell sequences and the reciprocation performed by the Borel-Laplace transform. This links together the compositional inversion with a multiplicative inversion, both “performed” by the Borel-Laplace transform weighting of the inverse function relation, and shows how reciprocal expressions related to a reciprocal Appell sequence pair are related to Hirzebruch’s criterion for the Todd class relation. (See a similar formulation in “Formal group laws and genera” by T. Panov at http://www.boma.mpim-bonn.mpg.de/data/30print.pdf.) The same two Appell sequences with the basic number sequences being the Bernoullis and the reciprocal integers are at the heart of the Euler-MaClaurin formalism.

On the other hand, performing the inversion by using the coefficients of the power series expansion of $h(x) = x/(e^x-1)$ leads to the formalism of the LIF of OEIS-[A134264][2] (see Example 3) with all sorts of related combinatorial structures, including those of noncrossing partitions and Dyck lattice paths, related to the Eulerian, Narayana, and Fuss-Catalan numbers among others. The LIF is used by Ardila, Rincon, and Williams to show the relation between cardinalities of connected and disconnected positroids. The partition polynomials themselves are a general Appell sequence that can be used to construct a “trajectory” through various classic number arrays and polynomials, with far reaching implications, I believe, one enticing fact being that the Eulerians and Narayanas are related to volumes of polytopes.

Almost forgot to mention that this skewed view of the arguments with the Appell umbral approach reveals some interesting structure. The Bernoulli polynomials (any Appell sequence) have the interesting property $(B.(0)+x)^n= B_n(x)$, so with our notation $B_n=B_n(0)$ and

$\displaystyle \left [e^{B.\;\omega} \right ]^{n+1}=exp\left [ (B.(0) + B.(0) +\; \cdots\; + B.(0) \right)\omega ],$

giving the coefficient of $\omega^n$ as

$\displaystyle [ B.(0) + B.(0) +\; \cdots \; + B.(0)) ]^n = B_n(B.(B.(B.(\cdots B.(0)))))=(-1)^n n!$

with $n+1$ summands on the left and $n+1$ umbral substitutions on the right.

The L.H.S. can be expanded as a multinomial treating each umbra as independent and not evaluating them until all monomial summands of degree $n$ have been formed. When that is done, the superscripts can be dropped to subscripts. This is equal to the R.H.S. which is iterated umbral substitution. For example, for $n=2$,

$\displaystyle (a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc)=3B_2+6(B_1)^2=2$

and

$\displaystyle (B.(B.(B.(0))))^2=B_2(B.(B.(0)))= 1/6 - (B.(B.(0)))^1 + (B.(B.(0)))^2=1/6-B_1(B.(0))+B_2(B.(0))=1/6- [B_1(0) -1/2] + 1/6 - B_1(0) + B_2(0)=2.$

For $n=3$, the result is $4 \; B_3 + 36 \; B_1 B_2 + 24 \; B_{1}^3=- 3!$ . The explicit expressions for these umbral reductions of the homogeneous [monomial symmetric polynomials][3], which hold for any Appell sequence, are those of the LIF of OEIS-[A248120][4], with its various combinatorial interpretations.

The operator formalism for umbral sustitution brings out clearly the relation between exponentiation of the base e.g.f. for the Bernoulli numbers and the iterated umbral composition. For example,

$\displaystyle e^{B.(B.(x))t}=e^{B.(x)D_{y=0}}e^{B.(y)D_{z=0}}e^{zt}=e^{B.(x)D_{y=0}}e^{B.(y)t}=e^{B.(x)D_{y=0}} \frac{t}{e^t-1} \; e^{yt}= \frac{t}{e^t-1} e^{B.(x)t}=(\frac{t}{e^t-1})^2 e^{xt} \; .$

These arguments reveal associations among the symmetric polynomials, convolutions, umbral compositions, and umbra, and shed light on the nature of the Hirzebruch criterion and why algebraically the Bernoulli sequence is the only sequence to satisfy it.

Returning to the formal Borel-Laplace transform arguments above, I should mention that the series $\sum_{n \ge 0} n! \; (-z)^n$ is the iconic divergent series related to the special function called the exponential integral, which is typically used to illustrate asymptotic series. The limit of the derivatives of the initial integral above as z approaches zero from the right are indeed equal to $(-1)^n (n!)^2$ even though the Taylor series is divergent for any nonzero value of z. The arguments can be formally extended to generating series expressed in terms of general indeterminates to give valid general Lagrange inversion formulas, as in my notes “Lagrange à la Lah”.

Some related stuff:

1) “Umbral presentations for polynomial sequences” by B. Taylor

2) “MOPS: Multivariable Orthogonal Polynomials (symbolically)” by Dumitriu, Edelman, and Shuman

3) “Groups and Lie algebras corresponding to the Yang-Baxter equation” by Bartholdi, Enriquez, Etingof, and Rains

4) “Multiplicative functions on the non-crossing partitions and free convolution” by R. Speicher (The relations between the moments and free cumulants through the Lagrange inversion formula in the paper is parallel to that of an Appell umbral compositional inverse pair with a Cauchy transform replacing the Laplace transform. See the entry on cumulants.)

5) Norlund published in 1924  in Vorlesungen uber Differenzenrechnung the formula (in umbral notation here) for the falling factorial $(x-1)_n = (x+\hat{B}.)^n=\hat{B}_n(x)$ where $e^{\hat{B}.(x)t}= ( \frac{t}{e^t-1})^{n+1} e^{xt}$. In his book on algebraic geometry, Hirzebruch notes Norlund’s work.

6) (Added Jan 18, 2017) “Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model)” by Scott and Sokal, see eqn. 3.1 on pg. 10 and eqn. 3.62 on pg. 24.