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


I'm afraid this is a bit vague (as I haven't done any studying for
about six months, and consequently mathematics is a bit of a blur!),
but the occasion last year when I stumbled across the formula for
coefficients of Chebyshev polynomials of the first kind that Denis
Feldmann mentioned in another thread today was one on which, seem-
ingly out of the blue (I can't decipher my own notes!), I 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)!

which I conjectured to be true for all real non-zero t, and all
real x with absolute value <= 1. I did some numerical experiments
(including some with complex values of t, and complex values of x
in the closed unit circle), and mumbled something about perhaps
proving that both sides satisfy the same second-order differential
equation (in x, I imagine), but I don't seem to have done any
serious work on trying to prove anything.

Does anyone know if any result along these lines - i.e. the same
equation, but with conditions on t and x stated more precisely,
and the limiting case t = 0 treated properly (it seems to work
for all real x with |x| < 1) - is true? (If so, I might force
myself to try to think about it again. Got to start studying
again somehow!) 8-P

--
Angus Rodgers
.

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