Summation involving 2F2 hypergeometric function

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 $$n=1,2,\ldots$$, $$z>0$$, $$0, and $$(x)_k=\Gamma(x+k)/\Gamma(x)$$ which is the Pochhammer symbol.

The inner sum is expressible through the $${}_2F_{2}$$ hypergeometric function. The following code

f[n_] := Sum[(z)^i/(n - i)!*1/(
Pochhammer[1 + a, i]*Pochhammer[1 - a, i])*
Sum[Pochhammer[-1 + a, j] Pochhammer[-1 - a, j]*(-z)^-j/j!, {j, 0,
i}], {i, 0, n}];

f[k] // FullSimplify


gives the expression that involves the $${}_2F_{2}$$ function.

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?

• Are you sure that this is a question about Mathematica the software? It seems to be entirely about the underlying math as you posed it. – MarcoB Feb 19 '19 at 3:33
• As @MarcoB suggests, maybe try math.stackexchange.com – Roman Feb 19 '19 at 4:44
• Do you have to have a simple algebraic expression for your sum? Because using your existing code, I get reasonable simple expressions when I plug integer values into your f[n] – Bill Watts Feb 20 '19 at 0:54
• @BillWatts f[n] is a polynomial in z of degree n, so it’s not that bad. I was hoping to express f as a single hypergeometric function though. – Alex Feb 20 '19 at 1:44