The two-parameter Wright function <http://dlmf.nist.gov/10.46> is defined as the infinite series
 $$
\phi (\alpha, \beta \, | z)=\sum\limits\_{k=0}^\infty \frac{z^k}{\Gamma(k+1) \Gamma(\alpha k + \beta) }
$$
It arises naturally as a solution of certain classes of differential equations in mathematical physics.


I would like to ask for formulas or bibliography about the representation of the function with elementary functions.


PS: The question has been asked on Math StackExchange where it got no traction:


<https://math.stackexchange.com/questions/2443604/representation-of-the-wright-function>



representation of the Wright function
Representations of $\phi(\alpha,\beta|z)$ in terms of more elementary functions exist only in special cases, see [arXiv:1703.01912](https://arxiv.org/abs/1703.01912) (2017) and [An Extension of Wright Function and Its Properties](https://www.hindawi.com/journals/jmath/2015/950728/) (2015):


$$\phi(1,\nu+1|-z^2/4)=2z^{-\nu}J\_\nu(z)$$
$$\phi(1,\nu+1|z^2/4)=2z^{-\nu}I\_\nu(z)$$
$$\phi(-1/2,1|-z)=1-\text{erf}\,(z/2)$$
$$\phi(-1/2,1/2|-z)=\pi^{-1/2}\exp(-z^2/4)$$


