On the non-commutative neutrix product of the distributions x(+)(lambda) and x(+)(mu)

Abstract

Let f and g be distributions and let g(n) = (g * delta(n))(x), where delta(n)(x) is a certain sequence converging to the Dirac delta function. The non-commutative neutrix product f circle g of f and g is defined to be the limit of the sequence {fg(n)}, provided its limit h exists in the sense that

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for all functions p in D. It is proved that

(x(+)(lambda)ln(p)x(+)) circle (x(+)(mu)ln(q)x(+)) = x(+)(lambda+mu)ln(p+q)x(+), (x(-)(lambda)ln(p)x(-)) circle (x(-)mu ln(q)x(-)) = x(-)(lambda+mu)ln(p+q)x(-),

for lambda + mu < -1; lambda,mu,lambda+mu not equal -1,-2,... and p,q = 0,1,2.....