### A new characteristic property of Mittag-Leffler functions and fractional cosine functions

A new characteristic property of the Mittag-Leffler function ${E}_{\alpha}\left(a{t}^{\alpha}\right)$ with 1 < α < 2 is deduced. Motivated by this property, a new notion, named α-order cosine function, is developed. It is proved that an α-order cosine function is associated with a solution operator of an α-order abstract Cauchy problem. Consequently, an α-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique α-order cosine function.