LAUSR

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