I am interested in power lines of form $$ f (z) = sum_ {k = 0} ^ infty frac {z ^ k} {(k!) ^ alpha}. $$ When $ alpha = 1 $is this $ exp (z) $, To the $ alpha = 2 $ This is a Bessel function and for a larger integer $ alpha $ We get a hypergeometric series. These special functions ($ alpha> 1 $) have integral expressions in the form of an integral of an elementary function.

Can be said of something for non-integral values $ alpha $? If $ alpha> 1 $ is there any real number, is there any hope to write an integer expression for the sum? Has this form of series been studied anywhere in the literature in general? Any useful techniques to work with?