On defining the distributions δkδk and (δ′)k(δ′)k by fractional derivatives

How to define products and powers of distributions is a difficult and not completely understood problem, and has been investigated from several points of views since Schwartz established the theory of distributions around 1950. Many fields, such as differential equations or quantum mechanics, require such operations. In this paper, we use Caputo fractional derivatives and the following generalized Taylor’s formula for 0<α<10<α<1ϕ(t)=∑i=0mCDˆ0,tiαϕ(0)Γ(iα+1)tiα+CDˆ0,t(m+1)αϕ(ζ)Γ((m+1)α+1)t(m+1)αto give meaning to the distributions δk(x)δk(x) and (δ′)k(x)(δ′)k(x) for all k∈Rk∈R. These can be regarded as powers of Dirac delta functions and have applications to quantum theory. At the end of this paper, the distributions logδ(t)logδ(t) and δ(t2)δ(t2) are given by the δ-sequence and the neutrix limit.