I am working on several problems that involve infinite sums, and the
expressions can get quite complicated. Sage can solve some of these, such as
m,p = var('m,p')
#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)
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)
gives
-1/4*sqrt(2)*sqrt(pi)*x^(3/2)
but the answer should be sinh(x). For other sums, Sage simply repeats what I
told it.
sum(x^(3*k)/factorial(2*k),k,0,oo)
I understand that Sage has limited exploitation of Maxima's hypergeometric
functionality, and I suspect this is the main issue. Are there any conceivable
workarounds?
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