On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions

Let F be a distribution in D[superscript 1] and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {F[subscript n](f(x))} is equal to h(x), where F[subscript n](x)= F(x) ∗ δ[subscript n](x) for n = 1, 2, . . . and {δ[subscript n](x)} is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ([superscript rs-1])((tanh x[subscript +])[superscript 1/r]) exists and δ([superscript rs-1])((tanh x[subscript +])[superscript 1/r]) = ∑[superscript s-1, subscript k=0] ∑[superscript K[subscript k], subscript i=0] ((-1)[superscript k]c[subscript s-2i-1,k] (rs)!/2sk!)δ([superscript k])(x) for r,s = 1, 2, . . ., where K[subscript k] is the integer part of (s-k-1)/2 and the constants c[subscript j,k] are defined by the expansion (tanh[superscript -1]x)superscript k = {∑[superscript ∞, subscript i=0] (x[superscript 2i+1]/(2i + 1))}[superscript k] = ∑[superscript ∞, subscript j=k] c[subscript j,k]x[superscript j], for k = 0,1,2,.... Further results also provided.