The n-Category Café

Misc. comments:

You might be interested in “Interactions between Lie theory and algebraic geometry” by Shilin Yu, tying together the GRR and the BCH formula.

The Todd op (mod signs) and its inverse are usually presented as differential ops, but, in a new MathOverflow question (not well received), I present the Todd op and its inverse also in terms purely of the finite difference op, which makes a direct connection to the Helmut Hasse formula for the Hurwitz zeta function and the Bernoulli function that Milnor presented in one of his papers. (The finite diff rep actually allows one of his divergent series derived from the diff op rep and related to the Harer-Zagier formula to be derived as a convergent series, but it is the asymptotic series that is desired.)

The Hurwitz $\zeta(s,z)$ is related to the Bernoulli function via the derivative $B_s(z)=-\partial_z \zeta(-s,z) = -s\zeta(-s+1,z)$, which evaluates as the Bernoulli polynomial $B_n(x)$ for a positive integer $n=s$. Alternatively, a Mellin transform gives the same result. The Hurwitz zeta and the Barnes zeta function (which can be expanded in terms of the Hurwtiz zeta) are related to the Casimir effect, Bose-Eisntein condensation, and solutions for the Schrodinger equation of atoms in a harmonic oscillator potential in “Basic zeta functions and some applications in physics” by Klaus Kirsten.

There are other interesting number arrays and associated combinatorics related to the deformed Todd operator variant of $\frac{1}{1-te^x}$ (see OEIS A131758 for a ref). A Taylor series expansion about $x=0$ gives rational functions with the Eulerian polynomials (see OEIS A008292 & A173018) in the variable $t$ in the numerator, and, when these are expanded as a power series, the powers of the integers appear; e.g., for $x^4/4!$ the coefficient is $(t^4+11t^3+11t^2+ t)/ (1-t)^5 = t + 2^4 t^2 + 3^4 t^3 + 5^4 t^4 + \cdots$. The Eulerian polynomials are related to a slew of other number arrays and functions important in analysis and number theory as noted in the OEIS entries above, including, of course, the Bose-Einstein and Fermi-Dirac distributions.

The definition of the Bernoulli numbers that sticks in my head is the one in Pierre Cartier’s paper Mathemagics: they are defined by the equation

$$B^n = (B + 1)^n$$

for $n \geq 2$, and by $B^0 = 1$. Then you change superscripts to subscripts.

This means the following. We start with $B_0 = B^0 = 1$. Then the equation above with $n = 2$ says

$$B^2 = B^2 + 2B^1 + 1.$$

Cancelling gives

$$0 = 2B^1 + 1$$

and so $B_1 = B^1 = -1/2$.

Next, take $n = 3$. The equation above says

$$B^3 = B^3 + 3B^2 + 3B^1 + 1.$$

Cancelling and changing superscripts to subscripts gives

$$0 = 3B_2 + 3B_1 + 1.$$

Then substituting in the value $B_1 = -1/2$ that we just computed gives $B_2 = 1/6$.

“And so on”. Clearly this is formal nonsense, but it’s formal nonsense that works.

Let’s look at some really simple umbral Sheffer calculus (courtesy of Blissard–the gnarly stuff was used by Sylvester and Cayley in their explorations of invariant theory). Let the umbral substitution op and its action be given by a differential op as

$$(a.)^n = S_{x \to a.} x^n = e^{a.\partial_{x=0}}x^n$$

$$= \sum_{k \geq 0} (a.)^k \frac{\partial_{x=0}^k}{k!} x^n = \sum_{k \geq 0} a_k \frac{\partial_{x=0}^k}{k!} x^n = a_n.$$

Then we can easily rigorously derive a fundamental identity of the finite difference calculus

$$(a.)^n = a_n = S_{x \to a.} x^n = e^{a.\partial_{x=0}}x^n = e^{a.\partial_{x=0}}(1-(1-x))^n$$

$$= e^{a.\partial_{x=0}} \sum_{k=0}^n (-1)^k\binom{n}{k} \sum_{j=0}^k (-1)^j \binom{k}{j}x^j$$

$$=\sum_{k=0}^n (-1)^k\binom{n}{k} \sum_{j=0}^k (-1)^j \binom{k}{j} a_j= (1-(1-a.))^n,$$

and this suggests another useful rep for umbral substitution

$$a_n = (a.)^n = S_{x \to a.} x^n = e^{-(1-a.)\partial_{x=1}}x^n = (1-(1-a.))^n$$

that can lead to interpretation for complex $s$ of

$$(a.)^s = a_s = (1-(1-a.))^s =e^{-(1-a.)\partial_{x=1}}x^s$$

$$= \sum_{k \geq 0} (-1)^k\binom{s}{k} \sum_{j=0}^k (-1)^j \binom{k}{j} a_j$$

as a Newton series, which is intimately related to Mellin transform interpolation and its analytic continuation. (With $a_j = (x+j)^{\alpha}$ and $s = b.$, the Bernoulli number umbra, the Newton series gives a variant of the Helmut Hasse formula for the Hurwitz zeta function, the Bernoulli function $B_\alpha(x) = -\alpha \; \zeta(-\alpha+1,x)$ of Milnor.)

The sequence of Bernoulli polynomials is an iconic Appell-Sheffer polynomial sequence (ASPS). Given any sequence of real or complex numbers, say $a_n$, with $a_0 =1$, an ASPS can be defined by umbral translation, or binomial convolution, via

$$A_n(x) = T_{x \to (x+a.)} x^n = (a.+x)^n = e^{a.\partial_x}x^n$$

$$=\sum_{k=0}^n \binom{n}{k} a_k x^{n-k} .$$

The Bernoulli polynomials and numbers $B_n(x)$ and $b_n$ can be generated by

$$B_n(x) = (b. +x)^n = e^{b.\partial_x} x^n = \frac{\partial_x}{e^{\partial_x}-1} x^n,$$

so the associated e.g.f.s are

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

Then

$$e^{-B.(1)t} = e^{-(b.+1)t} = e^{-b.t}e^{-t} = \frac{-t}{e^{-t}-1}e^{-t} =\frac{t}{e^{t}-1} = e^{b.t},$$

so

$$(-B.(1))^n = (-1)^n (b.+1)^n = (b.)^n = b_n,$$

the recursion relation noted by Cartier in Tom Leinster’s comments.

In these ways, the Bernoulli-Hirzebruch-Todd operator arises naturally in the umbral Sheffer operator calculus along with the associated inverse ops and the raising ops, which also involve the Bernoulli numbers, or more accurately the Riemann zeta numbers $\zeta(-n)$.

Of course, there are direct formulas for the Bernoulli numbers involving the Stirling numbers of the second kind or the face counts of the permutahedra, which are easily derived from the umbral Sheffer calculus, and these you find in toric topology and soliton solutions of the KdV equation and associated quadratic Riccati equation. In fact, from my formulas in OEIS A008292 on the Eulerian polynomials, you can discern a relation of the Bernoulli numbers to a formal group law and generalized Hirzebruch-Todd classes as John discusses. Expand as a Taylor series the shifted reciprocal of the bivariate e.g.f. for the Eulerian polynomials A(x,a,b)= (e^(ax)-e^(bx))/(ae^(bx)-be^(ax)), that is , x * (ae^(b x)-be^(a x)) / (e^(a x)-e^(b x)), using Wolfram Alpha and you will find the Bernoulli numbers. See the links adjacent to the OEIS formulas for refs on the associated generalized cohomology theory and toric topology (related to the convex polytopes the associahedra, permutahedra, and stellahedra).

For a deeper investigation of the applications of umbral calculus to topology, see “Symbolic calculus: A 19th century approach to MU and BP” by Nigel Ray in Homotopy Theory: Proceedings of the Durham Symposium 1983, London Mmathematical Society Lecture Notes Series, 117.

See the two MSE questions “What’s umbral calculus about?” and “Question about “baffling” umbral calculus result” and the MO-Q “How are Sheffer polynomials related to Lie theory?” for more introductory info and links.

The relationships among the Newton series of the finite difference calculus; the compositional inverse pair $h(x) = e^x-1$ and $h^{-1}(x)=\ln(1+x)$; the umbral inverse pair of binomial Sheffer Stirling polynomials of the second and first kinds, a.k.a. the Bell/Touchard/exponential and the falling factorial polynomials–core sequences in combinatorics; and the Appell Sheffer Bernoulli polynomials and their umbral inverse polynomials are perhaps most easily and elegantly revealed with the umbral Sheffer calculus. I can post a new short note on these in my web blog if you are interested. They reflect the relations among the basic Lie group operations of translation and multiplicative and compositional inversions.

https://mathoverflow.net/questions/10667/

The question that now jumps out at me: why should Temperature be like Integrating? (Or coolness (see Baez on What is Entropy)) like Differentiating?)