generating functions - Solving recurrence equation with indexes from negative infinity to positive infinity

In many cases, the recurrence equations that people are solving involves index of only non-negative values. Here I have a recurrence equation which arises from transport of light in an infinite 1D chain:

$a_m=sum _{j=1}^{infty } left(T_ja_{m+j}+T_ja_{m-j}right) + delta _{m,0}$

where $delta_{m,0}$ is the Kronecker delta function. i.e.:

$delta_{i,j} = begin{cases} & 1 text{ if } i=j \ & 0 text{ if } i neq j end{cases}$

Here I would like to solve $a_m$, where the index of m is from negative infinity to positive infinity, while $T_j$ is a given sequence, and p is just a given constant.

Defining the generating function $G(z)=sum _{k=-infty }^{infty } a_kz^k$, I found that:

$G(z)=frac{1}{1-sum _{k=1}^{infty } t_kleft(z^{-k}+z^kright)}$

The problem is, how am I going to do series expansion on G? Doing a simple expansion of $frac{1}{1-sum _{k=1}^{infty } t_kleft(z^{-k}+z^kright)}=sum _{j=0}^{infty } left(sum _{k=1}^{infty } t_kleft(z^{-k}+z^kright)right){}^j$ won't help. Since the power is too difficult to expand out.

And contour integration isn't helping as well, since it is too difficult to compute analytically or numerically too.

Here I would like to ask about direction in obtaining analytical solution, or approximated one.

And in my case, my function G is given by:

$G(z)=left(1+frac{3i}{2r^3}left(r^2left(ln left(1-frac{e^{i r}}{z}right)+ln left(1-e^{i r}zright)right)right)-i rleft(text{Li}_2left(frac{e^{i r}}{z}right)+text{Li}_2left(e^{i r}zright)right)+text{Li}_3left(frac{e^{i r}}{z}right)+text{Li}_3left(e^{i r}zright)right){}^{-1}$

p.s.:I have posted the same problem in Voofie.

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