representation of the Wright function

Question

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

Answer 1

Representations of $\phi(\alpha,\beta|z)$ in terms of more elementary functions exist only in special cases, see arXiv:1703.01912 (2017) and An Extension of Wright Function and Its Properties (2015):

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