Working on a problem of lattice Green functions (LGF) I encountered a summation shown in the title question: $$ \sum_{n=-\infty}^\infty I_{n+m}(a) I_{n-m}(b) I_{p-n}(c) . $$ It is part of wider summation: $$ \sum_{n=-\infty}^\infty \sum_{m=-\infty}^\infty I_{n+m}(a) I_{n-m}(b) I_{p-n}(c) I_{q+m}(c) $$ where all the indices n, m, p, q are integers. The above formula is part of still wider integrals: $$ \int_0^\infty dt \exp(-E t)\sum_n \sum_m I_{n+m}(a t) I_{n-m}(b t) I_{p-n}(c t) I_{q+m}(c t) $$ or $$ \int_0^\infty dt \exp(-E t) I_w(f t) \sum_n \sum_m I_{n+m}(a t) I_{n-m}(b t) I_{p-n}(c t) I_{q+m}(c t) $$ the former is LGF in 2 dimensions (2d) the latter - in 3d.
Does any one know, how to simplify any of these formulas getting rid of summations? Or how (IF) it could be done? There is Neumann/Graf addition formula which does that for the product of TWO Bessel functions but has any of you seen an analogue for three? How to intersperse the third index into summation? Any simplified versions for $a=b$ or $p=0$ or $q=0$ or $p=0=q$ or with $J_n$ instead of $I_n$ would be also of use. I tried various formulas from Gradshtein-Ryzhik and Abramovitz-Stegun but to no avail. Any hint how to attack the problem?