Application of fractional calculus to differential equations of Hermite's type (Q1075479)

In an earlier paper [Ann. Sc. Norm. Super. Pisa, Cl. Sci., III. Ser. 15, 1-24 (1961; Zbl 0099.314)] the author discussed the calculus of Holmgren- Riesz operators. These are integro-differential operators dependent on a complex parameter \(\alpha\) which for \(-\alpha =n\), \(n\in {\mathbb{N}}\) coincide with the operators of n-fold differentiation. In the present paper the second order differential equation \(V''-(2\delta x+\mu)V'+\delta \alpha V=0\) (\(\alpha\),\(\delta\),\(\mu\in {\mathbb{C}})\), which generalizes Hermite's differential equation, is reduced to an equivalent Holmgren-Riesz operator equation. The latter equation is solved in an obvious manner: one of the solutions is expressed by a generalized Rodriguez formula which also leads to a power series expansion. Some additional properties are obtained in the special case \(\alpha =n\in {\mathbb{N}}\) which corresponds to Hermite's equation in a strict sense.