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