On 2014-04-23, Karl S <art.live...@gmail.com> wrote:
> #taylor coefficient for erf(3x)
> a_erf(m) = (3)^(2*m+1)*(-1)^m*2/sqrt(pi)/(factorial(m)*(2*m+1))
>
> #coefficient of chebyshev polynomial
> c_erf_cheb(p) = sum(a_erf(m)*binomial(2*m+1,m-p)*4^-m,m,p,oo).simplify_full
> ()
>
> Here the function c_erf_cheb(p) ends up being
>
> -6/11*(bessel_I(6, -9/2) - bessel_I(5, -9/2))*sqrt(e)*e^(-5)/sqrt(pi)
Hmm, what is p in that result? For small values of p I get something
similar but not the same.
> which, to me, is a very useful answer. But other sums are simply wrong.
>
> k = var('k')
> sum(x^(2*k)/factorial(2*k),k,0,oo)
I'm working with Maxima 5.33.0. I get
simplify_sum ('sum(x^(2*k)/factorial(2*k),k,0,inf));
=> sqrt(%pi)*bessel_i(-1/2,x)*sqrt(x)/sqrt(2)
which seems to be cosh(x).
> sum(x^(3*k)/factorial(2*k),k,0,oo)
I get
simplify_sum ('sum(x^(3*k)/factorial(2*k),k,0,inf));
=> sqrt(%pi)*bessel_i(-1/2,x^(3/2))*x^(3/4)/sqrt(2)
Hope this helps,
Robert Dodier
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