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

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) * δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function (x). The function $\cosh _ + ^{ - 1}\left( {x + 1} \right)$ is defined by$$\cosh _ + ^{ - 1}\left( {x + 1} \right) = H\left( x \right){\cosh ^{ - 1}}\left( {\left| x \right| + 1} \right),$$where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition ${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right]$] exists and$${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right] = \sum\limits_{k = 0}^{s - 1} {\sum\limits_{j = 0}^{kr + r - 1} {\sum\limits_{i = 0}^j {{{{{( - 1)}^{kr + r + s - j - 1}}r} \over {{2^{j + 2}}}}\left( {\matrix{{kr + r - 1} \cr j \cr } } \right)} } } \left( {\matrix{j \cr i \cr } } \right)\left[ {{{\left( {j - 2i + 1} \right)}^s} - {{\left( {i - 2i - 1} \right)}^s}} \right]{\delta ^{(k)}}(x),$$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].

On the neutrix composition of x−−slnmx− and ln(1 + x+)

The neutrix composition [Formula: see text], [Formula: see text] is a distribution and [Formula: see text] is a locally summable function, is defined as the neutrix limit of the sequence [Formula: see text], where [Formula: see text] and [Formula: see text] is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function [Formula: see text]. The neutrix composition of the distributions [Formula: see text] and [Formula: see text] is evaluated for [Formula: see text] Further related results are also deduced.

The composition of the distributions x−λlnsx− and x+−r/λ

The composition [Formula: see text] of a distribution [Formula: see text] and a locally summable function [Formula: see text] is defined as the neutrix limit of the regular sequence [Formula: see text] In this paper, we prove that the neutrix composition of the distributions [Formula: see text] and [Formula: see text] exists and equals [Formula: see text] for [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the Beta function, [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text]

Generalized the q-digamma and the q-polygamma functions via neutrices

The q-digamma function ?q(x) and the q-polygamma functions ?(r)q(x), r ? N = {1,2,...} are defined for all x > 0 and 0 < q < 1. In this paper, the neutrices and the neutrix limit are used to define the q-digamma function q(x) and the q-polygamma functions ?(r)q(x), r ? N for all x ? R. Moreover, further results are given.

Some equalities on q-gamma and q-digamma functions

In this paper, we give some equalities on q-gamma and q-digamma functions for negative integer values of x by aid of using the concepts of neutrix and neutrix limit.

Some generalized equalities for the q-gamma function

The q-analogue of the gamma function is defined by ?q (x) for x > 0, 0 < q < 1. In this work the neutrix and neutrix limit are used to obtain some equalities of the q-gamma function for all real values of x.

On the composition of the distributions x-s+ lnmx+ and xμ+

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {Fn(f)}, where Fn(x) = F(x)*?n(x) and {?n(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function ?(x). The composition of the distributions x-s + lnm x+ and x? + is proved to exist and be equal to ?mx-s? + lnm x+ for ? > 0 and s,m = 1, 2,....