Finding a Closed Expression for an Infinite Sum with Positive Variables

[al2521300]

1. I want to find a close expression for the following infinite sum

$$\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n^2+x_1 y}}-\frac{1}{\sqrt{n^2+x_2 y}}\right)$$

Both $$x_i$$ and $$y$$ are greater than 0.

The Attempt at a Solution

I haven't got much really. I realize each sum independently is divergent, but together they would be convergent. I just don't know how to extract the convergent part into a closed expression. I try doing a power expansion for y around 0 and reach

$$\sum_{n=0}^{\infty}-\frac{\sqrt{\pi}Zeta(1+2n)(x_1^n-x_2^n)y^n}{\Gamma(\frac{1}{2}-n)\Gamma(1+n)n!}$$

but this is not much of an improvement over the original expression. Any help will be appreciated.

 

[lippyka]

This sum is not expressible in terms of standard mathematical functions. However, it can be expressed in terms of special functions such as the Hurwitz zeta function, the hypergeometric function, or the generalized hypergeometric function.