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1d
comment Geometric interpretation of the half-derivative?
The non-locality/locality has nothing to do directly with the gamma function. Look at the Cauchy integral for derivatives, which is equivalent to taking the finite part of the real convolutional integral representing the derivative. The dichotomy is related to branch cuts vs. poles. The gamma fct gives the correct residues for the poles to directly give the derivatives, a proportionality constant that is otherwise not necessary. And, the integral rep of the beta fct. (an. cont.) makes it a rep for x=1 of $D_x^{\alpha}\frac{x^{\beta}}{\beta!}=D_x^{-\beta}\frac{x^{-\alpha}}{(-\alpha)!}‌​$. Useful?
2d
revised The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)
Deleted comments on erroneous Wikipedia article.
Dec
12
comment An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
Perhaps Jack polynomials/functions play an important role. Anyway, I've pretty much figured out how the classic number arrays here, and others, enter the picture to my satisfaction. I believe the key actors are Appell sequences, non-crossing partitions, and certain methods of compositional inversion.
Dec
12
comment Why do Bernoulli numbers arise everywhere?
Generalized Bernoulli polynomials defined by powers of the e.g.f. (used in Hizebruch's criterion for the Todd class) have a rich history in the q-calculus as noted in Thomas Ernst's writings. Doyon, Lepowsky, Milas show how the Bernoulli polynomials are related to vertex operator algebras. The Bernoullis are related to the Euler, Eulerian, zizgag, Gennochi, ordered Bell numbers, and polylogarithms. Since the e.g.f. is tied so closely to exp, they occur in expansions of trig and hyperbolic functions. Quantum groups, solitons, and solns.of the KdV equation, I've noted elsewhere. Now shoes, ... .
Dec
12
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
mathoverflow.net/questions/6373/…
Dec
11
comment Why do Bernoulli numbers arise everywhere?
See also Vector Bundles and K-Theory by Allen Hatcher.
Dec
10
comment Why do Bernoulli numbers arise everywhere?
Basically, where ever you see the derivatve you could replace it with the Bernoulli polynomials and make some sense of it, but you are right, Ryan. I've never seen even one under my bed. Never seen an exponential there either. What was the OP thinking of?!
Dec
10
revised Why do Bernoulli numbers arise everywhere?
Misprints corrected
Dec
10
revised Why do Bernoulli numbers arise everywhere?
deleted 14969 characters in body
Dec
9
revised Why do Bernoulli numbers arise everywhere?
Added note. deleted some redundancie
Dec
9
revised A sum involving derivatives of Vandermonde
Added link. Deleted dead link.
Dec
8
revised Why do Bernoulli numbers arise everywhere?
Abbreviated some.
Dec
6
revised Why do Bernoulli numbers arise everywhere?
Added missing index. Important for the umbrally confused.
Dec
4
revised Why do Bernoulli numbers arise everywhere?
Improved a critical formula fixed link
Dec
1
revised What does the generating function $x/(1 - e^{-x})$ count?
more references
Nov
30
awarded  Necromancer
Nov
29
revised Why do Bernoulli numbers arise everywhere?
Intro added and coth operator
Nov
28
revised What does the generating function $x/(1 - e^{-x})$ count?
Added relation to volumes
Nov
28
revised Why do Bernoulli numbers arise everywhere?
Corrected link
Nov
27
comment Why is there a connection between enumerative geometry and nonlinear waves?
Also see "Bernoulli numbers and solitons" by G Rzadkowski tandfonline.com/doi/pdf/10.1142/S1402925110000635 The connections of the Bernoulli numbers to topology are well-known.