## 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 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.]

First, define the Bernoulli polynomials as the Appell sequence, $(B.(0)+x)^n=B_n(x)$, such that,

$f(B.(x+1))-f(B.(x))={f}'(x)$

when convergent. Action on $f(x)=e^{xt}$ gives the e.g.f. since

$e^{B.(x+1)t}-e^{B.(x)t}=t\;e^{xt}$   implies

$e^{B.(x)t}(e^t-1)=t\;e^{xt}$    and

$e^{B.(x)t}=\frac{t}{e^t-1}e^{xt}.$

Action on $f(x)=-ln(1-xt)$ gives

$-ln\left [1-B.(x+1)t \right ]+ln\left [1-B.(x)t \right ]= ln\left [ \frac{1-B.(x)t}{1-(B.(x)+1)t} \right ]=\frac{d\left [ -ln(1-xt) \right ]}{dx}=\frac{t}{1-xt}\;,$

an iconic o.g.f., and using the special linear fractional (Mobius) transformation $L(t,x)=\frac{t}{1+xt}$, whose inverse in $t$ is $L(t,-x)$, this can be expressed succinctly as

$ln\left [ 1+L[t,-(B.(x)+1)] \right ]=ln\left [ \frac{1-B.(x)t}{1-(B.(x)+1)t} \right ]=L(t,-x)=\frac{t}{1-xt},$

with compositional inverse in $t$

$L[e^t-1,B.(x)+1]=\frac{e^t-1}{1+(B.(x)+1)(e^t-1)}=L(t,x)=\frac{t}{1+xt}.$

[[[Retraction Oct. 14 2016: The previous line of equalities is patently wrong in general in at least two respects!!!! First, only if the umbral variables are treated as  regular variables throughout the composition for inversion does the claim hold that the first equality is the inverse or, what amounts to the same calculation, if the umbral compositional inverse is used first to reduce the umbral variable to $x$. Second, the umbral expression does not evaluate to $L(t,x)$. However, it is certainly true that inserting the umbral compositional inverse of the Bernoulli polynomials $\hat{B}_n(x)$ for $x$ gives the true relations

$ln\left [ 1+L[t,-(B.( \hat{B}.(x))+1)] \right ]=ln\left [ \frac{1-B.(\hat{B}.(x))t}{1-(B.(\hat{B}.(x))+1)t} \right ]=L(t,-\hat{B}.(x))=\frac{t}{1-\hat{B}.(x)t}$

$=ln\left [ 1+L[t,-(x+1)] \right ]=ln\left [ \frac{1-xt}{1-(x+1)t} \right ]$

and that the compositional inverse (as originally stated correctly further below) is then

$L[e^t-1,x+1]=\frac{e^t-1}{1+(x+1)(e^t-1)}= \frac{t}{1+p.(x)t} \neq \frac{t}{1+\hat{B}(x)t},$

where $p_n(x)$ are the polynomials of A019538 each divided /normalized by $n!$ (cf. also A131689). Consequently, the expressions in red brackets below should be ignored and the first replaced with the blue. In addition, all references to the Hirzebruch criterion have been removed (being tangential in any event, a vestige of an earlier discussion).]]]

Together {{{they}}} the special linear  fractional tranformations comprise the formal group law

$FGL(y,z;x)=L[L(y,-x)+L(z,-x),x]=\frac{y+z-2x\;yz}{1-x^2yz},$

which (according to [Lenart and Zainoulline][1]) corresponds to the Euler characteristic.

For $x=0$, {{{$(B.(0))^n=B_n$ are the Bernoulli numbers, $(B.(0)+1)^n=(B.(1))^n=(-B.(0))^n=(-1)^nB_n$, and}}} the FGL specializes to

$FGL(y,z;0)=y+z,$

the fundamental additive FGL associated with the infinitesimal generator $d/dt=D_t$ and the iterated op $(D_t)^n$ with action of translation $exp(x\;D_t)f(t)=f(t+x)$.

For $x=-1$, this specializes to

$FGL(y,z;-1)=\frac{y+z+2yz}{1-yz},$

the self-dual Lah FGL  associated to the infinitesimal generator $(1+t)^2\frac{d}{dt}$, and, with a shift in coordinates, to the iterated op $(t^2D_t)^n=t^{n}Lah(:tD_t:)$, related to the Lah polynomials OEIS-A111596 (A094638), with the action $exp(x\;t^2D_t)f(t)=f(t/(1-xt))$, the special linear fractional transformation. Here, by definition, $(:tD_t:)^n=t^nD_t^n$.

More generally for indeterminates $a_n$ with $a_0=1$, action on $f(a.,t)=-ln(1-a.\;t)$

gives (precisely when convergent and formally usefully otherwise)

$ln\left [ \frac{1-B.(a.)t}{1-(B.(a.)+1)t} \right ]=\frac{t}{1-a.t}=\sum_{n \ge 0} a_n t^{n+1},$

an o.g.f. for the series, which itself can be extended as an Appell sequence defined by the o.g.f.

$\boldsymbol{O}_A(t,x)=\frac{t}{1-(a.+x)t}=\sum_{n \ge 0} (a.+x)^n t^{n+1}=\sum_{n \ge 0} A_n(x) t^{n+1}.$

Letting $a_n=\hat{B}_n(x)$, the umbral compositional inverse for the Bernoulli polynomials, i.e., $B_n(\hat{B}.(x))=x^n=\hat{B}_n(B.(x))$, we get

$\boldsymbol{O}_{\hat{B}}(t,x)=ln[ \frac{1-B.(\hat{B}.(x))t}{1-(B.(\hat{B}.(x))+1)t}]=ln[ \frac{1-xt}{1-(x+1)t}]$

$=\sum_{n \ge 0} \frac{(x+1)^{n+1}-x^{n+1}}{n+1} t^{n+1},$

and, consistently,

$\hat{B}_n(B.(x))=\frac{(B.(x)+1)^{n+1}-(B.(x))^{n+1}}{n+1}=\frac{d}{dx}\;\frac{x^{n+1}}{n+1}=x^n.$

The compositional inverse {{{, through the same substitution above,}}} is

$\boldsymbol{O}_{\hat{B}}^{(-1)}(t,x)=\frac{e^t-1}{1+(x+1)(e^t-1)}.$

[This is essentially the equation 11.1 (2) on page 94 of F. Hirzebruch’s Topological Methods in Algebraic Geometry for $R(y;x)$ associated with the generalized Todd class. A slight change of coordinates is needed. Compare the expression here with my Sept. 18, 2014, formula E in OEIS A008292 and the formula in Hirzebruch’s book (English translation, 2’nd corrected printing of the third edition, 1978). It is presented again on pg. 12 of his 2007 paper “Eulerian Polynomials”.]

Together they comprise the formal group law

$FGL(y,z;x)=\boldsymbol{O}_{\hat {B}}^{(-1)}[\boldsymbol{O}_{\hat{B}}(y,x)+\boldsymbol{O}_{\hat{B}}(z,x),x]=\frac{y+z-(1+2x)\;y z}{1-x(1+x)\;yz}.$

For $x=-1$,

$\boldsymbol{O}_{\hat{B}}^{(-1)}(t,-1)=e^t-1$,    $\boldsymbol{O}_{\hat{B}}(t,-1)=ln(1+t),$

and

$FGL(y,z;-1)=y+z+yz,$

the multiplicative FGL associated with the Todd genus and the infinitesimal generator $(1+t)\frac{d}{dt}$ at the identity related by a coordinate shift to the iterated op. $(t\frac{d}{dt})^n=\phi_n(:tD_t:)$, the Bell or Stirling polynomials of the second kind, with action $exp(x\;tD_t)f(t)=f(e^xt)$, a dilation.

For $x=-1/2$,

$FGL(y,z;-1/2)=\frac{y+z}{1+\;yz/4},$

the Lorentz group FGL, related to the Atiyah-Singer signature ([Lenart and Zainoulline][2], “Towards generalized cohomology Schubert calculus via formal root polynomials”. Hirzebruch also states the relation to tanh, whose Taylor series contains the Bernoullis.)

So a dance between the Bernoullis and the elegant reciprocal polynomials, their umbral compositional inverses, is weaving a path through some basic formal groups, associated genera, and the conformal Lie algebra $sl_2(C)$ and associated group, the conformal global subgroup of the Witt Lie (and Virasoro) algebra.

There’s a relation to elliptic functions, and more, as discussed by L & Z, and the entry Bernoulli Appells here presents connections to solitons, Ricatti equations, the KdV equation, and vector fields. For infinite (and finite) matrix reps related to the Pascal matrix, see the notes on Infinigens.

Note: The FGLs here encode local Lie action and are connected to diverse combinatoric and geometric structures related to compositional inversion. With $g(z)=\frac{1}{{f}'(z)}$, $\omega=f(z)$, and $f^{(-1)}(\omega)=z$,

$\exp(x\;g(z)D_z)\;z= \exp(x\;D_{\omega})\;f^{(-1)}(\omega)=f^{(-1)}(x+\omega)=f^{(-1)}(x+f(z)),$

and so

$\exp(f(y)\;g(z)D_z)\;z=f^{(-1)}(f(y)+f(z))=FGL(y,z).$

With $f(0)=0$, satisfaction of the algebraic, axiomatic definitions for FGLs is transparent along with the assertion that all FGLs are isomorphic with the additive FGL corresponding to the action of simple translation, for the complex field at least.

See OEIS-[A145271][3] for one relation among many to combinatorics (refined Eulerian integers) and for the typical relations to Lie groups, including vector fields and autonomous ODEs. Key relations can be extended to non-analytic power series on a graded basis. See “[Formal Groups and Applications][4]” by M. Hazewinkel and “[Formal group laws and genera][5]” by T. Panov for general introductions.

Tying the FGL for the Todd class, the operational definition of the Bernoullis, and the umbral compositional inverse together:

The compositional inverse is generated from the exponentiated infinitesimal generator above acting on $z$ evaluated at $z=0$, and the infinigen for the Todd FGL is given by $g(t)D_t$ where

$1/g(t)=\frac{d(\ln(1+t))}{dt}=1/(1+t)=\ln[1+B.(1+t)]-\ln[1+B.(t)]$

$=\sum_{n \geq 0} (-1)^{n} \frac{B.(1+t)^{n+1}-B.(t)^{n+1}}{n+1}=\frac{1}{1+\hat{B}.(B.(t))},$

where $\hat{B}_n(x)$  as specified above are the umbral compositional inverses (reciprocal polynomials) of the Bernoulli polynomials, so the character of the Bernoulli polynomials (and their elegant escorts, the reciprocal polynomials), the Todd FGL, and Lagrange inversion (and Lie theory and combinatorics) are inextricably interlinked, as could be anticipated from the action of the Bernoulli polynomials as derivations and the connection  of derivations to the inverse function theorem and therefore FGLs.

(Added Oct. 17, 2016) Note the Stieltjes / Hilbert / Cauchy transform

$\int_x^{x+1} \frac{1}{z-t} \; dt = ln[\frac{1-x/z}{1-(1+x)/z}]$

and that the analysis in the entry “Appell polynomials,  cumulants, …” can be applied:

$\boldsymbol{O}_{\hat{B}}^{(-1)}(t,x)=\frac{e^t-1}{1+(x+1)(e^t-1)} =\frac{t}{1 \; + \; RT(t)}$,

giving the R transform as essentially the e.g.f. of the Bernoulli numbers

$RT(t) = -1 \; +\; (1+x) \; t + \; \frac{t}{e^t-1}= -1 + (1+x) \; t+e^{B.(0)t}$.

Consequently, the Lagrange inversion formula related to non-crossing partitions / Dyck paths A134264 can be used to generate $\boldsymbol{O}_{\hat{B}}(t,x)$ from

$RT(t) + 1 = \frac{t}{\boldsymbol{O}_{\hat{B}}^{(-1)}(t,x)}=1 \; + \;(x+1/2) \; t \; + \;B_2(0) \; t^2/2! \; + \; B_4(0) \; t^4/4! \; + \cdots .$

with the Bernoulli numbers as weights. Variation of $x$ changes only the weight of the singleton component of the partitions (or steps) designated $h_1$.

For $x = -1/2$,

$\boldsymbol{O}_{\hat{B}}^{(-1)}(t,-1/2)= 2 \; \tanh(t/2) = \sum_{n > 0} 4 \; (2^{n+1}-1) \; \frac{B_{n+1}}{n+1} \; \frac{t^n}{n!}.$

Compare these coefficients with A000182 (also see MO-Q: What does the generating function $x / (1-e^{-x})$ count?).

Related stuff:

1) “The signature theorem and some of its applications” by L. Nicolaescu.

2) “Cobordism theory and the signature theorem” by C. Rovi (slide presentation)

3) My question on Math Overflow and links therein “The Riemann zeta at positve integers and an Appell sequence related to fractional calculus” https://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials

4) “Characteristic classes of symmetric products of quasi-projective varieties” by Cappel, Maxim, Schurmann, Shaneson, and Yokura

5) “Caculation of Hirzebruch genera for manifolds …” by T. Panov

6) “Hirzebruch genera and manifolds with torus action” by T. Panov

7) “Torus actions and their applications in topology and combinatorics” by Buchstaber and Panov

8) “Toric genera” by Buchstaber, Panov, and Ray

9) “Toric topology” by Buchstaber and Panov

10) “Elliptic formal group laws, integral Hirzebruch genera and Krichever genera” by Buchstaber and Bunkova

11) The action of iterated infinitesimal generators (Lie derivatives / vector fields) were graphically modeled by Cayley with rooted trees in the 1850s and related by Comtet to special polynomial sequences (and much earlier by Scherk). This is related to a normal ordering of the iterated derivative ops and their relations to Sheffer sequences. Another approach formalizing these actions as algebras of pre-Lie or post-Lie quadratic operads and binary trees has been exploited by Chapoton, Loday, Vallette and others in which  the generating series of this entry and the others on the Bernoulli polynomials naturally occur.

12) “Formal groups, Witt vectors and free probability” by Friedrich and McKay

13) “Formal groups and their role in the apparatus of algebraic topology” by Buchstaber, Mishchenko, and Novikov

14) “Life and work of Friedrich Hirzebruch” by Zagier