On the Non-Commutative Neutrix Product of the Distributions xλ + and xμ +

Abstract

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 (xλ + lnp x+) ◦ (xμ + lnq x+) = xλ+μ + lnp+q x+, (xλ − lnp x−) ◦ (xμ − lnq x−) = xλ+μ − lnp+q x−, for λ + μ < −1; λ, μ, λ + μ = −1, −2,... and p, q = 0, 1, 2.... .