>
>
> 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?
>
>
This actually works out of the box in the latest development release, 
probably with the same fix in #16224.

sage: sum(x^(3*k)/factorial(2*k),k,0,oo)
sqrt(pi)*x^(3/4)*sqrt(1/(pi*x^(3/2)))*cosh(x^(3/2)) 
sage:  sum(x^(3*k)/factorial(2*k),k,0,oo).simplify_radical()
cosh(x^(3/2))


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