Hypergeometric extension to deal with double summation

Question

I would like to be able to find an expression for summations of the form $$ \sum_{l,m=1}^{\infty}\frac{x^{l}y^{m}}{a^{2}+l^{2}+m^{2}}, $$ at least in terms of special functions, although I can't even see how to do this. I'm aware of hypergeometric functions that deal with the single summation version but nothing that deals with the double summation. Thanks in advance for any help.