explicit formula for $ _2F_2(1,1;2;2;z) $

Question

Is it an explicit formula for $$ _2F_2(1,1;2;2;z) ,$$ where $$_2F_2(a,b;c;d;z)=\sum_{n\geq 0}\frac{(a)_n(b)_n}{(c)_n(d)_n n!}z^n .$$

thanks you in advance

Answer 1

$$ \begin{align} _2F_2(1,1;2;2;z) &=\sum_{n=0}^\infty\frac{n!n!}{(n+1)!(n+1)!}\frac{z^n}{n!}\\ &=\sum_{n=0}^\infty\frac{z^n}{(n+1)(n+1)!}\\ &=\frac1z\int_0^z\sum_{n=0}^\infty\frac{t^n}{(n+1)!}\mathrm{d}t\\ &=\frac1z\int_0^z\frac{e^t-1}t\,\mathrm{d}t \end{align} $$ This doesn't have a closed form without using special functions like the Exponential Integral or Incomplete Gamma.