I'm interested in a transformation of the form $$ {_3F_3}\left({2,a,a\atop 1,1+a,1+a};z\right)=e^z{_3F_3}\left({?,?,?\atop ?,?,?};-z\right). $$ Many times, hypergeometric functions of the form ${_nF_n}(z)$ can have a transformation applied to write them in the form of $e^z{_nF_n}(-z)$. Kummer's transformation is a good example. We have $$ {_1F_1}\left({a\atop b};z\right)=e^z{_1F_1}\left({b-a\atop b};-z\right). $$ Examples for ${_2F_2}$ an $_3F_3$ can also be found, for example here. I have been unable to find the transformation I'm interested anywhere. Is there a systematic procedure for finding what I'm looking for?
Edit: Here is an example of a full derivation for a special $_2F_2$ function.
Edit2:
Using DLMF 16.5.1 we can write $$ {_3F_3}\left({2,a,a\atop 1,1+a,1+a};z\right)=a^2\int_0^1\int_0^1(tu)^{a-1}e^{ztu}(1+ztu)\,\mathrm dt\mathrm du. $$ This allows us to express the ${_3F_3}$ as a sum of ${_2F_2}$'s as $$ {_3F_3}\left({2,a,a\atop 1,1+a,1+a};z\right)={_2F_2}\left({a,a\atop 1+a,1+a};z\right)+z\frac{a^2}{(a+1)^2}{_2F_2}\left({1+a,1+a\atop 2+a,2+a};z\right). $$ Now in terms of ${_2F_2}$ there may be a chance to tease out the $e^z$ term.