bio  website  tcjpn.wordpress.com 

location  Japan  
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visits  member for  3 years, 11 months 
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1d

comment 
Geometric interpretation of the halfderivative?
The nonlocality/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 
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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, noncrossing partitions, and certain methods of compositional inversion. 
Dec 12 
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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 qcalculus 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 
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Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
mathoverflow.net/questions/6373/… 
Dec 11 
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Why do Bernoulli numbers arise everywhere?
See also Vector Bundles and KTheory by Allen Hatcher. 
Dec 10 
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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 
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Why do Bernoulli numbers arise everywhere?
Misprints corrected 
Dec 10 
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Why do Bernoulli numbers arise everywhere?
deleted 14969 characters in body 
Dec 9 
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Why do Bernoulli numbers arise everywhere?
Added note. deleted some redundancie 
Dec 9 
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A sum involving derivatives of Vandermonde
Added link. Deleted dead link. 
Dec 8 
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Why do Bernoulli numbers arise everywhere?
Abbreviated some. 
Dec 6 
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Why do Bernoulli numbers arise everywhere?
Added missing index. Important for the umbrally confused. 
Dec 4 
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Why do Bernoulli numbers arise everywhere?
Improved a critical formula fixed link 
Dec 1 
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What does the generating function $x/(1  e^{x})$ count?
more references 
Nov 30 
awarded  Necromancer 
Nov 29 
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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 wellknown. 