Re: Apropos of Chebyshev polys: t^{-1}*sin(t*sin^{-1}(x))?



Angus Rodgers wrote:
On Sat, 30 May 2009 20:13:28 -0700 (PDT), galathaea
<galathaea@xxxxxxxxx> wrote:

On May 28, 9:04 am, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:

[...]

t^{-1}*sin( t*sin^{-1}(x) )

oo { j } x^{2*j+1}
= sum { prod [(2*i-1)^2 - t^2] } * ---------
j=0 { i=1 } (2*j+1)!

[...]

this is one of the "cauchy formulas"
(i know... very descriptive name...)

there is a similar one for the T tchebyshef

the book by louis comtet "advanced combinatorics"
in chapter 3 exercise 29
gives as a reference an article by teixera
" sur le developpement de x^k en serie ordonnee
suivant les puissances de sinus "
in nouvelles annales v15 (1896) 270-4

if you expand it
it can be reduced to a direct application
of the definition of bell polynomials

That Comtet book came up a few years ago as the great reference
for something else I was interested in, and I've been meaning to
buy it ever since (not that I ever get around to reading anything
these days), but it's awfully expensive! Even the cheapest used
copy at amazon.co.uk is more than 90 pounds (more than the price
of a new copy from Springer, which is 75 pounds). Oh well, some
day! Thanks for the reference. (I used to find this kind of
thing fun, and might do again.)

--
Angus Rodgers

Angus,

Great work! Exciting stuff. Can you help me with a proof, too?

Here read this (do not think about the letters as words but math
variables, then the qy_-it):

My name is Martin Musatov. I have been doing some experimentation with
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findings as they relate computational complexity to a mathematical
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.

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