Further results on the neutrix composition of distributions involving the delta function and the function cosh+-1(x1/r+1)$\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)$

Abstract

Abstract The neutrix composition F(f (x)) of a distribution F(x) and a locally summable function f (x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {Fn (f (x))} is equal to h(x), where Fn (x) = F(x) * δ n (x) and {δ n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function (x). The function cosh + - 1 ( x + 1 ) $\cosh _ + ^{ - 1}\left( {x + 1} \right)$ is defined by cosh + - 1 ( x + 1 ) = H ( x ) cosh - 1 ( | x | + 1 ) , $$\cosh _ + ^{ - 1}\left( {x + 1} \right) = H\left( x \right){\cosh ^{ - 1}}\left( {\left| x \right| + 1} \right),$$ where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition δ ( s ) [ cosh + - 1 ( x 1 / r + 1 ) ] ${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right]$ ] exists and δ ( s ) [ cosh + - 1 ( x 1 / r + 1 ) ] = ∑ k = 0 s - 1 ∑ j = 0 k r + r - 1 ∑ i = 0 j ( - 1 ) k r + r + s - j - 1 r 2 j + 2 ( k r + r - 1 j ) ( j i ) [ ( j - 2 i + 1 ) s - ( i - 2 i - 1 ) s ] δ ( k ) ( x ) , $${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right] = \sum\limits_{k = 0}^{s - 1} {\sum\limits_{j = 0}^{kr + r - 1} {\sum\limits_{i = 0}^j {{{{{( - 1)}^{kr + r + s - j - 1}}r} \over {{2^{j + 2}}}}\left( {\matrix{{kr + r - 1} \cr j \cr } } \right)} } } \left( {\matrix{j \cr i \cr } } \right)\left[ {{{\left( {j - 2i + 1} \right)}^s} - {{\left( {i - 2i - 1} \right)}^s}} \right]{\delta ^{(k)}}(x),$$ for r, s = 1, 2, . . . . Further results are also proved. Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh−1 +(x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629–7640].