Two variants of fractional powers of Hankel-Clifford transformations of order ν ge; 0 with parameter α are introduced and studied their properties. The operational formulas are developed. We study pseudo-differential operators associated with the symbol a(x,y) and their integral representations. Some properties of Sobolev type spaces are studied. Fractional powers of Hankel-Clifford transformations are used in the solution of partial differential equation involving Kepinski type operators Δ*1, ν, μ, α and Δ*2, ν, μ, α. © 2014 Springer Basel.