On Mellin transforms of solutions of differential equation $$\chi ^{(n)}(x)+\gamma _{n}x\chi (x)=0$$

Abstract

In this paper, for $$ n=2,3,\ldots ,$$ we consider the differential equation $$\begin{aligned} \chi ^{(n)}(x)+\gamma _{n}x\chi (x)=0,\quad \left\{ \begin{array}{ll} \gamma _{n}=(-1)^{k},&{}\quad n=2k,\\ \gamma _{n}=-1,&{}\quad n=2k+1, \end{array}\right. \end{aligned}$$ and find the linear independent solutions in terms of the higher-order Airy functions ( $$n=2k$$ ) and the higher-order Levy stable functions ( $$n=2k+1 $$ ). The integral representations of solutions are presented and their Mellin transforms are also given.