The aim of the paper is to present several new relationships involving the hyperharmonic function introduced by Mez$ddot{o}$ in (I. Mez$ddot{o}$, Analytic extension of hyperharmonic numbers, Online J. Anal. Comb. 4, 2009) which is an analytic extension of the hyperharmonic numbers. These relations are obtained by using some fractional calculus theorems as Leibniz rules and Taylor like series expansions.