summing over bessel function

Yes, as you say, something like

sage: G = lambda x,k: bessel_K(0,2*pi*(x+k))
sage: [sum([G(x,k) for k in [1..100]]) for x in [0,1,2,3]]
[0.000917807437286879, 1.22307638250860e-6, 1.87088814696602e-9, 3.03065972802175e-12]

However,

Sometimes bessel_K(nu,z) is denoted K_nu(z) in the literature. In
PARI, nu can be complex and z must be real and positive.

So we get errors trying to sum over negative numbers.

There is also the scipy algorithm.

sage: G = lambda x,k: bessel_K(0,2*pi*(x+k),algorithm='scipy')
sage: [sum([G(x,k) for k in [1..100]]) for x in [0,1,2,3]]
[0.000917807437286507, 1.22307638250709e-6, 1.87088814709167e-9, 3.03065985167335e-12]

But this also does not play well with negative input.

There is also the problem that bessel_k(0,0) is undefined, if you were going to do that.

You could try using Maxima inside Sage, which does give negative input meaning. But it won't do anything symbolic with a sum to infinity of Bessel K.

sage: maxima_console()
<snip>

(%i1) sum(bessel_k(0,1+k),k,-3,3); 

bessel_k: bessel_k(0,0) is undefined.
 -- an error. To debug this try: debugmode(true);
(%i2) sum(bessel_k(0,1.5+k),k,-3,3);
(%o2)              2.3647955753643517 - 8.514365020222318 %i
(%i3) sum(bessel_k(0,1.5+k),k,-100,100);
(%o3)           2.4596346751510154 - 1.2033957278916282e+42 %i
(%i4) sum(bessel_k(0,.01+k),k,-100,100);
(%o4)            5.894176649568293 - 5.299305601915854e+42 %i

Notice also the very large imaginary part, which I have no idea whether to trust.

The bessel_K function is only implemented as far as numerical evaluation (thanks to PARI). For example, if you try:

sage: x = var('x')
sage: f = bessel_K(0,x)
...
TypeError: Cannot evaluate symbolic expression to a numeric value.