I'm trying to simplify the following sum: $$ \sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!}, $$ where $n=1,2,\ldots$, $z>0$, $0<a<1$, and $(x)_k=\Gamma(x+k)/\Gamma(x)$ is the Pochhammer symbol.
The inner sum is expressible through the ${}_2F_{2}$ hypergeometric function of the following form: ${}_2F_{2}[1,-i;2-a-i,2+a-i;z]$. The latter is a terminating series, so the entire double sum is a polynomial in $z$ of degree $n$.
My question is to what extent is the entire double sum above can be simplified? For example, can it be reduced to a single hypergeometric function of some sort?
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