A non‐commutative neutrix product of distributions

Let f and g be distributions and let gn = (g ∗ δn)(x), where δn(x) is a certain sequence converging to the Dirac delta function. The non-commutative neutrix product f ◦g of f and g is defined to be the limit of the sequence {fgn}, provided its limit h exists in the sense that N−lim n→∞ 〈f(x)gn(x), φ(x)〉 = 〈h(x), φ(x)〉, for all functions φ in D . It is proved that (x+ ln p x+) ◦ (xμ+ ln x+) = x + ln x+, (x− ln x−) ◦ (xμ− ln x−) = x − ln x−, for λ + μ < −1; λ, μ, λ + μ = −1,−2, . . . and p, q = 0, 1, 2. . . . .