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 Yearling
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asked Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?
Jan
26
comment What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$?
You might find this informative physics.stackexchange.com/questions/21595/…
Jan
23
comment Alternative definition of the Lagrange Inversion formula
@IraGessel, this might be a good intro to the historiy of the method, not sure if it includes anything about Abel though: books.google.com/…. I typed in the title and Graves to see a google books excerpt
Jan
20
comment Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$
Real integrals such as those for the gamma function and Riemann zeta (standard Mellin transforms for Re(s) > 0) can be expanded to complex integrals using the Hankel contour, just as Riemann did in his famous paper on the Reimann zeta. The same can be done for the beta integral, which is a particularly simple Mellin transform at the foundations of one class of fractional calculus that has been developed any number of ways and for which Euler developed his iconic integral for the gamma function. Which contour integral are you really looking for?
Jan
12
awarded  Yearling
Jan
9
comment What does the generating function $x/(1 - e^{-x})$ count?
See also Kirillov, "Two more variations on the triangular theme" (p. 12 and 13).
Jan
9
comment Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?
See also Kirillov, "Two more variations on the triangular theme" (p. 12 and 13).
Jan
8
revised Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$?
Corrected initialization of summation and added full series.
Jan
5
comment Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$?
@Mathlover, no thanks, MO is best taken in small doses, can be toxic when cut with the wrong fillers (^ o).
Jan
5
comment Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$?
The question has been answered already with reference to work of grade A mathematicians, so what is the point to closing it?
Jan
5
comment Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$?
@Mathlover, may I suggest that you simply rewrite the question to ask about the utility of and perennial interest, in the last 100 years, in asymptotic expansions of the upper incomplete gamma function connected below to the lower series and in the Mittag-Leffler function for the upper series. Several recent books and papers addressing the topic are available. Perhaps some other users on this venue have some pertinent knowledge. Qtherwise, consider math stackexchange--users there have broader interests in general and can be less rigid in nature.
Jan
4
comment Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$?
Recent overview: "Divergent series: tamig the tails" by Berry and Howls (michaelberryphysics.files.wordpress.com/2013/06/berry482.pdf)
Jan
4
answered Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$?
Dec
8
comment Connection between the Chebyshev polynomials and the Faber polynomials
Follow the OEIS links for relations to walks on paths of P_n and K_n graphs.
Dec
8
comment Inversion, Koszul duality, combinatorics and geometry
Perhaps item 2 is addressed by Drakes' thesis "An inversion theorem for labelled trees ..." people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf.
Dec
8
comment What does the generating function $x/(1 - e^{-x})$ count?
See also the trees described on page 16 of Drakes' thesis "An inversion theorem for labelled trees ,,," people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf enumerated by atanh(x) = x - x^3/3 + x^5/5 - ... and tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ...,
Dec
8
comment Compositional inversion and generating functions in algebraic geometry
To the degree that the inverse pair can be represented by certain types of trees, Drakes' thesis "An inversion theorem for labelled trees ..." people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf has some insights.
Dec
6
revised Connection between the Chebyshev polynomials and the Faber polynomials
Revamped
Dec
5
revised Connection between the Chebyshev polynomials and the Faber polynomials
Added missing square roots.
Dec
4
comment Connection between the Chebyshev polynomials and the Faber polynomials
See A263916 for other connections of the Faber polynomials to the Chebyshev polynomials.