The convolution of functions and distributions

The non-commutative convolution f∗g of two distributions f and g in D′ is defined to be the limit of the sequence {(fτn)∗g}, provided the limit exists, where {τn} is a certain sequence of functions in D converging to 1. It is proved that |x|λ∗(sgnx|x|μ)=2sin(λπ/2)cos(μπ/2)sin[(λ+μ)π/2]B(λ+1,μ+1)sgnx|x|λ+μ+1, for −1<λ+μ<0 and λ,μ≠−1,−2,…, where B denotes the Beta function.