Resonance

Posted on February 20, 2025 by Tom Copeland

I have several hundreds of pages of math notes I need to organize and publish here, but given current events and the plights of so many millions affected by the current callous regime in the U.S., I’m compelled by my conscience to direct people to a poignant speech that reaches out to people of kindred spirits from almost a century ago:

The Great Dictator Speech – Charlie Chaplin + Time – Hans Zimmer (INCEPTION Theme)

(https://youtu.be/w8HdOHrc3OQ?si=7FITT6LCfKdMRy9N)

Posted in Uncategorized | Leave a comment

Boil and bubble: Conjuring the digamma and log(D) by umbrally mixing the logarithm with the Bernoulli function

Posted on December 31, 2024 by Tom Copeland

Define the logarithm of the derivative operator, \log(D), or more precisely \ln(D), as the infinitesimal generator (IG) for the fractional integroderivative op D^t of the Heaviside-Pincherle family of the fractional calculus such that, for x,s,t real,

\displaystyle e^{t \; \ln(D_x)} H(x) \frac{x^s}{s!} = D^t H(x) \frac{x^s}{s!} = H(x) \frac{x^{s-t}}{(s-t)!},

where H(x) is the Heaviside step function and

\displaystyle H(x) \frac{x^{-n}}{(-n)!} = \delta^{(n-1)}(x) = D^{n}H(x)

with \delta(x) as the Heaviside-Dirac delta function / operator. Alternatively, to circumvent issues with prematurely evaluating \frac{x^{-n-1}}{(-n-1)!} as 0 use the limit

\displaystyle H(x) \frac{x^{-n}}{(-n)!} := \lim_{a \to 0} \frac{1}{2} H(x) \left[\frac{x^{-n+a}}{(-n+a)!} + \frac{x^{-n-a}}{(-n-a)!}\right]

and

\displaystyle D^t \frac{x^{s}}{s!} = \lim_{a \to 0} D^t \frac{1}{2} H(x) \left[\frac{x^{s+a}}{(s+a)!} + \frac{x^{s-a}}{(s-a)!}\right] = \lim_{a \to 0} e^{t \ln(D)} \frac{1}{2} H(x) \left[\frac{x^{s+a}}{(s+a)!} + \frac{x^{s-a}}{(s-a)!}\right].

There are other methods to get around this apparent pitfall, such as to parse the computations as

D^t H(x) \frac{x^{-n}}{(-n)!}= D^t D^n H(x) = e^{t\;\ln(D)}e^{n\;\ln(D)} H(x)

= H(x) e^{(t+n)\;\ln(D)}1= H(x) \frac{x^{-t-n}}{(-t-n)!}

= e^{n\;\ln(D)}e^{t\;\ln(D)} H(x) = D^n D^t H(x) = D^n H(x) \frac{x^{-t}}{(-t)!} = e^{n\ln(D)}H(x) \frac{x^{-t}}{(-t)!}.

(See the links in my answer to and in the sidebar of the MO-Q “What’s the matrix of logarithm of derivative operator ln(D)? What is the role of this operator in various math fields?” for links to several of my older deeper notes on this family of fractional integroderivatives and the associated infinitesimal generator \ln(D). I also have an upcoming blog post updating this topic.)

Ultimately the differential operator rep for the IG can then be expressed from the calculus of the lowering and raising operators of Appell Sheffer polynomial sequences (and other methods) as

\displaystyle \ln(D) = -\ln(x) + \psi(1+xD) = -\ln(x) + \ln(B.(1+xD)),

where

\psi(z) = \partial_z \ln((z-1)!)

is the digamma function and

B_s(z) =-s \zeta(1-s,z) = -\partial_z \zeta(1-s,z)

is the Bernoulli function defined in terms of the Hurwitz zeta function.

The following analysis unpacks and establishes the umbral compositional / umbral translation identity

\psi(z) = T_{z \to (z+b.) = (B.(z)}\ln(z) = \ln(B.(z)) = \ln(z+b.),

giving the digamma function as the umbral polynomial composition of the natural logarithm with the Bernoulli function (z+b.)^s := (B.(z))^s := B_s(z), or, equivalently, as the umbral number translation of the logarithm with the Bernoulli numbers (b.)^n = b_n.

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Posted in Math | Tagged Appell Sheffer polynomial sequences, Bernouilli numbers, Bernouilli polynomials, Bernoulli function, binomial Sheffer polynomial sequences, Compositional inverse, Compositional inversion, Differential operators, Digamma function, Fractional integroderivative operators, Heaviside-Pincherle fractional calculus, Hurwitz zeta function, infinitesimal generator, ln(D), log(D), Logarithm of the derivative operator, normal ordering, Polygamma functions, Reciprocal functions, Umbral calculus, Umbral inverse composition, Umbral shift operator, Umbral translation operator | Leave a comment

Bernoulli number umbral translation / polynomial umbral substitution operator and binomial Sheffer polynomial sequences

Posted on December 19, 2024 by Tom Copeland

The Helmut Hasse finite diff rep of Bernoulli number umbral translation, or Bernoulli polynomial umbral substitution, discussed in my previous posts,

\displaystyle T_{x \to (x+b.) = B.(x)}f(x) = f(b.+x) = f(B.(x))

\displaystyle = \sum_{n \geq 0} (-1)^n \binom{b.}{n} \sum_{k =0}^n (-1)^n \binom{n}{k} f(k+x)

\displaystyle = \sum_{n \geq 0} \frac{1}{n+1} \sum_{k =0}^n (-1)^n \binom{n}{k} f(k+x),

is most easily seen as an umbral generalization of Newton series translation, a translation expressed in the basis of falling factorials / generalized binomial coefficients rather than the divided power monomials of Taylor series.

Details, reprising earlier notes, of relations to the operator calculus of binomial Sheffer polynomial sequences are presented in my pdf

Bernoulli number umbral translation / polynomial umbral substitution and binomial Sheffer polynomial sequences.

Caveat: I let the equals symbol do a lot of work here. It would perhaps be best to use the symbols \leftrightharpoons and \rightleftharpoons with, e.g., e^{b.\partial_x} \rightleftharpoons e^{(b.)_{\underline{.}}\delta_x} with the upper lifted/raised tick on the right side of the symbol indicating the op on the RHS gives convergent results when acting on a wider domain (lifted increased width) of functions than the op on the LHS does and vice versa for the lower depressed tick (lowered decreased width) on the left side of the symbol.

Posted in Math | Tagged Bernoulli number, Bernoulli polynomials, binomial Sheffer polynomial sequences, Chu-Vandermonde identity, Compositional inverse, Falling factorials, Helmut Hasse formula, Hurwitz zeta function, Newton series, Operator calculus, Taylor series, Umbral calculus, Umbral substitution, Umbral translation | Leave a comment

The Joy of Polygammary: Umbral witchcraft and the polygamma functions vis a vis the Bernoulli function

Posted on December 11, 2024 by Tom Copeland

The objective of this short pdf is to reveal, via the extended umbral operational calculus of Appell Sheffer polynomial sequences, relationships among the Bernoulli function;  its umbral inverse function, the reciprocal function; and the digamma function and its derivatives, the polygamma functions:

The Joy of Polygammary: Umbral witchcraft and the polygamma functions vis a vis the Bernoulli function (pdf)

Related stuff:

“On polylogarithms, Hurwitz zeta functions, and the Kubert identities” by Milnor. Note that the umbral Bernoulli number translation op / Bernoulli polynomial subsitution op in my pdf above and earlier ones provides the correct extension of the Bernoulli diff op (Todd op)

\displaystyle T_{x \to (x+b.)=B.(x)} = B(\partial_x) = \frac{\partial_x}{e^{\partial_x}-1}

to

\displaystyle T_{x \to (x+b.)=B.(x)}=\sum_{n \geq 0} \frac{1}{n+1} \sum_{k=0}^n (-1)^k \binom{n}{k} T_{x \to x+k}

so that the correct dual identity for

\displaystyle \frac{e^{\partial_x}-1}{\partial_x} \gamma_1(x) = x \; \ln(x) - x

on page 314 of Milnor is

\displaystyle \gamma_1(x) = \ln\left( \frac{(x-1)!}{\sqrt{2 \pi}}\right) = \sum_{n \geq 0} \frac{1}{n+1} \sum_{k=0}^n (-1)^k \binom{n}{k} [(x+k) \; \ln(x+k) - (x+k)]

for \displaystyle x > 0 (see this Demos plot for a numerical check).

Taking the derivative w.r.t. x gives an expansion for the digamma function

\displaystyle \psi(x) = \frac{d}{dx}\ln\left(\left(x-1\right)!\right) = \sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{k=0}^{n}\left(-1\right)^{k}\ \frac{n!}{k!\left(n-k\right)!}\ln\left(x+k\right)

for x > 0 (see this Demos plot for a numerical check).

Iterating the differentiation generates the polygamma functions, and it is in the sense of the operation above that the umbral subsitution

\displaystyle \ln(B.(x)) := T_{x \to (x+b.) = B.(x)} \ln(x) = \psi(x)

can be interpreted, which for action on x^n gives B.(x)^n = B_n(x) = (b.+x)^n, involving the Bernoulli polynomials and numbers.

This is consistent with the operational definition of the Bernoulli number umbral translation / Bernoulli polynomial umbral substitution for a wide class of functions f(x)

\displaystyle f(B.(x+1)) - f(B.(x)) = f(b.+1+x) - f(b.+x) = \partial_x f(x)

and the recursion formula for the digamma function, that is,

\displaystyle \ln(B.(x+1)) - \ln(B.(x)) = \psi(x+1) - \psi(x) = \partial_x \ln(x) = \frac{1}{x}.

This is also consistent with an extension of the umbral inversion relationship between the Bernoulli and the reciprocal polynomials to that between the Bernuolli function and reciprocal function.

For the polynomials for n=0,1,2, \cdots,

\displaystyle B_n(x) = e^{b.\partial_x} x^n = (b.+x)^n = \sum_{k=0}^n \binom{n}{k} b_n x^{n-k} = T_{x \to (x+b.)=B.(x)} x^n

and

\displaystyle R_n(x) = e^{r.\partial_x} x^n = (r.+x)^n = \sum_{k=0}^n \binom{n}{k} \frac{1}{k+1} x^{n-k}

\displaystyle = T_{x \to (r.+x) = R.(x)} x^n = \int_{x}^{x+1}t^n dt = \frac{(x+1)^{n+1}-x^{n+1}}{n+1},

then

\displaystyle x^n = \frac{\partial_x}{e^{\partial_x}-1} \frac{e^{\partial_x}-1} {\partial_x}x^n = e^{b.\partial_x}e^{r.\partial_x} x^n = e^{b.\partial_x}R_n(x) = R_n(B.(x))

\displaystyle = e^{r.\partial_x}e^{b.\partial_x} x^n = e^{r.\partial_x}B_n(x) = B_n(R.(x)) ;

and for the functions for s real or complex, with the Hurwitz zeta function \zeta(s,x),

\displaystyle B_s(x) = (b.+x)^s := T_{x \to (x+b.)=B.(x)} x^s = -s \; \zeta(-s+1,x)

and

\displaystyle R_s(x) = (r.+x)^s := \int_{x}^{x+1}t^s dt = \frac{(x+1)^{s+1}-x^{s+1}}{s+1},

then

\displaystyle R_s(B.(x)) = x^s = B_s(R.(x)).

For the limiting case s \to -1,

\displaystyle R_{-1}(B.(x)) = \lim_{s \to -1} \frac{(B.(x)+1)^{s+1}-B.(x)^{s+1}}{s+1} =\ln(B.(x)+1) - \ln(B.(x) )

\displaystyle = \ln(B.(x+1)) - \ln(B.(x))= \partial_x \ln(x) = \frac{1}{x} = \psi(x+1)-\psi(x)

\displaystyle = \lim_{s \to -1} \frac{B_{s+1}(x+1)-B_{s+1}(x)}{s+1}= \lim_{s \to -1}[-\zeta(-s,x+1) + \zeta(-s,x)],

and

\displaystyle B_{-1}(R.(x)) = \zeta(2, R.(x)) = \sum_{n \geq 0} \frac{1}{(n + x)^2}|_{x \to R.(x)} = \sum_{n \geq 0} \frac{1}{(n + R.(x))^2}

\displaystyle = \sum_{n \geq 0} \frac{1}{(R.(x+n))^2} = \sum_{n \geq 0} R_{-2}(x+n)

\displaystyle = \sum_{n \geq 0} -[\frac{1}{x+n+1} - \frac{1}{x+n}] = \sum_{n \geq 0} \frac{1}{(x+n+1)(x+n)} = \frac{1}{x}.

This Desmos plot demonstrates the good fit for x > 0 of

\displaystyle y = \sum_{n=0}^{40}\frac{1}{n+1}\sum_{k=0}^{n}\left(-1\right)^{k}\ \frac{n!}{k!\left(n-k\right)!}\ln\left(\frac{x+1+k}{x+k}\right),

an approximation of \displaystyle R_{-1}(B.(x)) = T_{x \to (x+b.)=B.(x)}R_{-1}(x),

with \displaystyle y = \frac{1}{x}.

Added Dec. 30, 2024:

The set of slides “Fractional calculus and zeta functions” by Arran Fernandez reviews (with a brief bio) Jerry Keiper’s work (pdf frame 23) on the relationship of the Riemann Hurwitz zeta function to fractional derivatives of the digamma function in Keiper’s thesis (1975) “Fractional calculus and its relationship to the Riemann zeta function” and more current work by others on fractional derivatives of the Riemann zeta. Keiper provided a generalization of the the relationship I show above between derivatives of the digamma function and the Hurwitz zeta. Using a different type of fractional derivative than I use, Keiper showed, for Re(s) >1,

\displaystyle \zeta(s,x) = (-1)^s \frac{{}_{-\infty}D_x^{s-1}}{(s-1)!} \psi(x).

Then, with this type of fractional derivative, for Re(s) > 0,

\displaystyle B_{-s}(x) = s\zeta(1+s,x) = (-1)^{s+1} \frac{{}_{-\infty}D_x^{s}}{(s-1)!} \psi(x)

with the limiting case

\displaystyle B_{0}(x) = 1 = - \lim_{x \to 0^{+}} \frac{\psi(x)}{(x-1)!}.

Posted in Math | Tagged Appell- Sheffer polynomials, Bernoulli function, Bernoulli numbers, Bernoulli polynomials, Digamma fuction, Finite difference operator, Helmut Hasse formula, Hurwitz zeta function, operational calculus, Polygamma functions, Reciprocal function, Reciprocal polynomials, Riemann zeta function, Taylor series expansion, Umbral calculus, Umbral inverse, Umbral translation operators | Leave a comment

On Euler’s derivation of the e.g.f. of the Bernoulli numbers

Posted on October 8, 2024 by Tom Copeland

I added a supplement to one of my old MSE questions “Original author of an exponential generating function for the Bernoulli numbers?, that elaborates on Euler’s method of deriving the ecponential generating function of the Bernoulli numbers using a modern matrix format rather than Euler’s format. Hopefully, this clarifies Euler’s derivation for the reader.

Posted in Math | Tagged Bernouilli numbers, Divergent series summation, Helmut Hasse formula, Hurwitz zeta function, Laplace transform, Mellin- transform, Riemann zeta function | Leave a comment

The Thermodynamic 3-D Permutahedron

Posted on August 26, 2024 by Tom Copeland

I’ve written over several years on the permutahedra / permutohedra–a family of convex polytopes–and the connections among their geometry, in particular, their associated refined Euler characteristics / refined, signed face polynomials; the algebra of multiplicative inversion of formal Taylor series, or, exponential generating functions; a matrix calculus, a Lie infinitesimal generator calculus, and differential operator calculus associated with Appell Sheffer polynomial sequences; symmetric polynomial / function theory; and an infinite dihedral group interweaving mutiplicative and compositional inversions of formal Taylor and Laurent series. But only today have I come across the relation of the 3-dimensional permutahedron, a.k.a. the truncated octahedron, to the Legendre transformations in thermodynamics as explained in the following paper;

Thermodynamic Venn diagrams: Sorting out forces, fluxes, and Legendre transforms by W. C. Kerr and J. C. Macosko

Vignette: I knew of Ronald Fox’s work cited in the paper since he was one of my brilliant and entertaining professors eons ago with, as he was proud to boast, a justifiably “big ego” which he was fond of saying “was other people’s problem, not mine”. Yet, he told his class–I had skipped class that day– “It blew my mind” when I correctly answered for the first time, in his many years of teaching, a nonlinear homework problem in his preferred standard textbook on thermodynamics–the apparent answer he had accepted up to that time was actually wrong. A friend/classmate told me this and that was good for my ego and at the same time a humbling reminder that everyone makes mistakes. (It was through googling ‘Fox thermodynamics Legendre transformation’ that I came across the article above, so yet again he has taught me some interesting mathematical physics.)

Posted in Math | Tagged 3-dimensional permutahedron, Legendre transformation, thermodynamics, truncated octahedron | 2 Comments

Umbral witchcraft with Bernoulli and Blissard

Posted on August 25, 2024 by Tom Copeland

Galvanized by comments in the recent post Bernoulli Numbers and the Harmonic Oscillator by John Baez on the blog site The n-Category Café, I’ve decided to post a fraction of some sets of notes I’ve developed on the appearance of the Bernoulli numbers, polynomials, and function in diverse areas of mathematics. I find the umbral Sheffer calculus to often be the most comprehensive and elegant way to derive and view the various relationships among the formulas involved in the related concepts, so I pay homage to Blissard in the title who was one of the first to use umbral characters in deriving Bernoulli number identities.

First, I’ll reiterate the relationship between a formal group law and the Bernoulli numbers:

The refs attached to my formulas dated Sep 18 2014 in OEIS A008292 are relevant to the connections between formal group laws and the Bernoulli numbers. As I commented in the Café, the shifted reciprocal of the bivariate e.g.f. for the Eulerian polynomials, the Exp for the hyperbolic formal group law, generates the Bernoulli numbers. In more detail, given the compositional-inverse pair of functions 

Exp_{(Eulerian)}(x;a,b) = A(x,a,b) = \frac{e^{ax}-e^{bx}}{a e^{bx}-b e^{ax}}

= x + (a+b) \frac{x^2}{2!} + (a^2+4ab+b^2)\frac{x^3}{3!} + (a^3+11a^2b+11ab^2+b^3) \frac{x^4}{4!} + \cdots

and

Log_{(Eulerian)}(x;a,b) = B(x,a,b) = \frac{1}{a-b} \ln\left(\frac{1+ax}{1+bx}\right) 

= x - (a+b)\frac{x^2}{2} + (a^2+ab+b^2)\frac{x^3}{3} - (a^3+a^2b+ab^2+b^3)\frac{x^4}{4} + ... ,

the Bernoulli numbers appear in the shifted reciprocal 

\frac{x}{Exp_{(Eulerian)}(x;a,b)} = x \frac{a e^{bx}-b e^{ax}}{e^{ax}-e^{bx}}

= 1 + \frac{-1}{2}(a + b) x  + \frac{1}{6}(a - b)^2 \frac{x^2}{2!} + \frac{-1}{30} (a - b)^4 \frac{x^4}{4!} + \cdots .

Relevant refs are “Elliptic formal group laws, integral Hirzebruch genera and Krichever genera” by Victor Buchstaber and Elena Bunkova (the Bernoulli numbers occur on pgs. 36 & 37), “Towards generalized cohomology Schubert calculus via formal root polynomials” by Cristian Lenart and Kirill Zainoulline, and the more recent papers “Theta divisors and permutohedra” and “Chern-Dold character in complex cobordisms and theta divisors” by Buchstaber and A.P. Veselov (the Eulerian polynomials are the h-polynomials of the permutohedra).

A(x,a,b) satisfies

\frac{dA}{dx} = (1+aA)(1+bA),

a Ricatti equation, and can be written in terms of a Weierstrass elliptic function (see Buchstaber & Bunkova), so

A(B(z)+x) = e^{x(1+az)(1+bz)\partial_z} z

and the formal group law is

FGL(x,y) = A(B(x,a,b) + B(y,a,b),a,b) = \frac{x+y+(a+b)xy}{1-(ab)xy},

called the hyperbolic formal group law and related to a generalized cohomology theory by Lenart and Zainoulline.

Note (1+az)(1+bz)\partial_z = \partial_x + (a+b)z\partial_z + (ab)z^2\partial_x is a linear combination of the infinitesimal generators for SL_2.

In my pdf in my post “The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera” at my Web blog Shadows of Simplicity, I note in detail the relations of the compositional inverse pair  A(x,a,b) and B(x,a,b)  to the KdV equation and, therefore, to the Burgers’ equation, the heat equation, the Schwarzian derivative, and more.

The coefficients of \frac{x}{Exp_{(Eulerian)}(x;a,b)} containing the Bernoulli numbers can be used to generate Log_{(Eulerian)}(x;a,b) via the partition polynomials of OEIS A134264, the refined Narayana partition polynomials, enumerating / labeling Dyck paths, polygon dissections, forests, and noncrossing partitions; therefore, the Bernoulli numbers can be generated from Log_{(Eulerian)}(x;a,b) by the inverse set of partition polynomials of A350499. This pair of sets of partition polynomials allows conversions between the free cumulants and the free moments of free probability theory and between the coefficients of a compositional-inverse pair of Laurent series. (In fact, at the bottom of p. 11 of “Todd polynomials and Hirzebruch numbers” by Buchstaber and Veselov is an example of the use of A350499 to calculate the Bernoulli numbers, of which they were unaware until I notified them in June.)

For details of the relationship between the Hirzebruch-Todd criterion and the K-P equation mentioned in comments in the Café, see my pdf

The Khovanskii-Pukhliko formula, the Hirzebruch-Todd criterion, the Bernoulli op, and umbral witchcraft.

(See also the answers to the MO-Q “Hirzebruch’s motivation of the Todd class” and “Formal Groups, Witt vectors and Free Probability” by R. Friedrich and J. McKay.)

For the links between the Bernoulli polynomials, Newton series, and the derivative / tangent space of functions, see my pdf

Conjuring the continuous from the discrete with umbral witchcraft: the derivative and the Bernoulli function.

For a review of a natural extension of the Bernoulli polynomials to the Bernoulli function of Milnor, see my pdf

Résumé of the Bernoulli function.

(See also my answer to the MO-Q “Transformation converting power series to Bernoulli polynomial series” and my posts “The Kervaire-Milnor Formula and Bernoulli Numbers“, “Differintegral Ops and the Bernoulli and Reciprocal Polynomials“, and “Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion“.

For some relationships among the quantum zeta function, the quantum partition function, heat kernels, statistical mechanics, the harmonic oscillator, Hermite and Bernoulli polynomials, the Bernoulli function and Hurwitz zeta function, and Bose-Einstein condensates, see my pdf

(forthcoming)

Posted in Math | Tagged Bernouilli polynomials, Bernoulli function, Bernoulli numbers, Bernoulli operator, complete homogeneous symmetric polynomials, Compositional inverse, Elliptic genera, Eulerian polynomials, Formal group law, Hirzebruch criterion, Hirzebruch-Todd class / genus, Hyperbolic group law, Kdv equation, Multiplicative inverse, Reciprocal, Todd class, Todd operator, Umbral calculus, Umbral inverse | 3 Comments

A Schur Thing – Appendix to As Above, So Below: The Up-Down Operators for the (m)-Associahedra Partition Polynomials

Posted on August 24, 2023 by Tom Copeland

This is a pointer to the appendix

A Schur Thing – Appendix to As Above, So Below: The Up-Down Operators for the (m)-Associahedra Partition Polynomials (pdf)

to the earlier set of notes “As Above, So Below . . .”. (A link to this pdf is provided in that post as well.)

In these notes, a Lagrange-Schur-Jabotinsky identity relating different coefficients of a power series raised to different integer powers is used to prove the raising and lowering operations of the sets [A^{(m)}] of (m)-associahedra partition polynomials are related to the set [N] of noncrossing partition polynomials (both sets introduced in various previous posts) by

[N] [A^{(m)}] = [A^{(m)}][N]^{-1} = [A^{(m+1)}]

and

[N]^{-1} [A^{(m)}] = [A^{(m)}][N] = [A^{(m-1)}].

Consequently, the sets [N^{(m)}] of (m)-noncrossing partition polynomials, which satisfy the identities

[A^{(m)}] = [N^{(m)}][R] = [N^{(m)}][A^{(0)}],

also satisfy

[N^{(m)}] = [N]^m

for m any integer.

Posted in Math | Tagged Ladder operators, Lagrange inversion, Lagrange-Schur-Jabotinsky identity, lowering / annihilation/ destruction / down operations, m-Narayana polynomals, m-noncrossing partitions, raising / creation / up operations, Refined m-associahedra polynomials | Leave a comment

(m)-Associahedra and (m)-Noncrossing Partition Polynomials and the Infinite Dihedral Group

Posted on July 26, 2023 by Tom Copeland

I was playing around once again yesterday with the basic algebraic relations among the sets of (m)-associahedra partition polynomials [A^{(m)}] and the sets of (m)-noncrossing partitions polynomials [N^{(m)}] = [N]^m, which I’ve presented in several posts over the last year or so, and decided to post a few defining relations for the dual sets of polynomials in a question on Math Stack Exchange in the hope that the group would be recognized by someone. The MSE user Karl pointed out that it sounds like I was describing the infinite dihedral group, linking to the associated Wikipedia article. This led me to the Wiki on the dihedral group and a set of group relations that are shared by my group of partition polynomials. Each set [N]^m plays the role of a rotation r_m in the dihedral group and each set [A^{(m)}], a reflection s_m. In the following I’ll show the applicability of these relations under the substitution operation I’ve illustrated in previous posts.

An infinite group \mathcal{N} is formed by iterating the substitution operation on [N]^1 = [N] and its inverse [N]^{-1}. The elements of this infinite group are [N]^{m} where m is any integer; [N]^0=[I], the identity; and, e.g., [N]^{3} = [N][N][N] and [N]^{-3} = [N]^{-1}[N]^{-1}[N]^{-1} under the repeated operation.

The infinite sets of (m)-associahedra partition polynomials satisfy for any integer m

(I)

[A^{(m)}]^2 = [I]

and

(II)

[N]^{\pm 1}[A^{(m)}] = [A^{(m\pm1)}].

That is, [A^{(m)}] is involutive and [N] and [N]^{-1} are the ladder ops–the raising and lowering ops–for the infinite set \mathcal{A} comprised of the infinite sets [A^{(m)}], where m runs over the infinite set of integers, as well as the ladder ops for the group \mathcal{N}.

For any integers i and j, clearly, by the definition of \mathcal{N} above,

[N]^i[N]^j = [N]^{i+j},

and the relation

[N]^{\pm 1}[A^{(m)}] = [A^{(m\pm1)}]

implies

[N]^{i}[A^{(j)}] = [A^{(j+i)}],

or equivalently

[N]^{-i}[A^{(j)}] = [A^{(j-i)}].

Recalling [A^{(m)}]^2 = [I], so [A^{(m)}]^{-1} =[A^{(m)}], and taking the inverse of this last equality gives

([N]^{-i}[A^{(j)}])^{-1} = ([A^{(j-i)}])^{-1},

implying

[A^{(j)}] [N]^{i} = [A^{(j-i)}].

Similarly, since

[A^{(j)}] = [N]^j[A^{(0)}] = [A^{(j)}]^{-1}= ([N]^j[A^{(0)}])^{-1} = [A^{(0)}] [N]^{-j},

finally

[A^{(i)}][A^{(j)}] = [N]^i[A^{(0)}][N]^j[A^{(0)}]

= [N]^i[A^{(0)}][A^{(0)}][N]^{-j}

= [N]^i[N]^{-j} = [N]^{i-j}.

Consequently, with [N]^i \to r_i and [A^{(i)}] \to s_i, my group satisfies the four relations

r_i r_j = r_{i+j}, \; r_i s_j = s_{i+j}, \; s_i r_j = s_{i-j}, \; s_i s_j = r_{i-j}

presented in the Wikipedia article on the dihedral group.

The group also satisfies the conjugation relations

1) [A^{(m)}][N]^n[A^{(m)}] = [N]^{-n}

and

2) [A^{(0)}][A^{(m)}][A^{(0)}] = [A^{(-m)}].

An analogous algebraic realization of the infinite dihedral group can be obtained by scaling the indeterminates by starting with multiplicative and compositional inversion of exponential generating functions (Taylor series) rather than ordinary generating function (power series). The extrapolation in both cases includes Laurent series. It would be interesting if there were other multivariate realizations under substitution.

Related stuff:

“Noncrossing partitions under rotation and reflection” by Callan and Smiley (pg.6)

“The power of group generators and relations: an examination of the concept and its applications” by Zhou

The Geometry and Topology of the Coxeter Groups by Davis

Group Theory by Milne

The CRM Winter School on Coxeter groups

Generalized Dihedral Group at GroupProps

“Dihedral group” by K. Conrad

Posted in Math | Tagged (m)-associahedra partition polynomials, (m)-Narayana partition polynomials, (m)-noncrossing partitions polynomials, infinite dihedral group, reflections, rotations, Weyl-Coxeter group A_n | 1 Comment

Iterative formula for the Kreweras-Voiculescu polynomials–the noncrossing partitions polynomials

Posted on July 19, 2023 by Tom Copeland

The Kreweras-Voiculescu partition polynomials, which I usually call the noncrossing partitions or the refined Narayana partition polynomials of OEIS A134264–flag and enumerate distinct noncrossing partitions and give the free moments in terms of the free cumulants in free probability theory. They are often useful in mathematical / physical analyses involving compositional inversion and are also of broad use in combinatorics characterizing parking functions, trees, Dyck lattice paths, cluster complexes, and other combinatorial geometric constructs. I’ve derived yet another method for iteratively determining these polynomials in the pdf

Iterative formula for the Kreweras-Voiculescu polynomials–the noncrossing partitions polynomials

Posted in Math | Tagged Compositional inverse, Iteration, Iterative generator, Kreweras polynomials, Lagrange inversion, Noncrossing partitions, Voiculescu polynomials | 2 Comments