Note on the Distribution Composition $(x_+^\mu )^\lambda$

Abstract

The composition of the distributions $x^\lambda$ and $x_+^\mu$ is evaluated for $\lambda = -1,-2,\ldots \quad \mu >0$ and $\lambda \mu \in {{\mathbb {Z}}}^- .$ Further results are deduced.

1 Introduction

In the following, we let ${{\mathcal {D}}}$ be the space of infinitely differentiable functions with compact support, let ${{\mathcal {D}}}[a,b]$ be the space of infinitely differentiable functions with support contained in the interval [a, b], and let ${{\mathcal {D}}'}$ be the space of distributions defined on ${{\mathcal {D}}}$.

we define the locally summable function $x_+^\lambda ,$ for $\lambda >-1$, by

$$\begin{aligned} x_+ ^\lambda =\left\{ \begin{array}{cc} x^\lambda , &{}x>0,\\ 0, &{} x<0. \end{array} \right. \end{aligned}$$

The distribution $x_+^\lambda$ is then defined inductively for $\lambda <-1$ and $\lambda \ne -2,-3, \ldots$ by $(x_+^\lambda )' =\lambda x_+^{\lambda -1}.$ It follows that if s is a positive integer and $-s-1<\lambda < -s$, then

$$\begin{aligned} \langle x_+ ^\lambda ,\varphi (x) \rangle = \int _0^\infty x^\lambda \left[ \varphi (x)- \sum _{k=0} ^{s-1} {\varphi ^{(k)} (0)\over k!} x^k\right] dx \end{aligned}$$

for arbitrary $\varphi$ in ${{\mathcal {D}}}$. In particular, if $\varphi$ has its support contained in the interval $[-1,1]$, then

$$\begin{aligned} \langle x_+ ^\lambda ,\varphi (x) \rangle = \int _0^1 x^\lambda \left[ \varphi (x)- \sum _{k=0} ^{s-1} {\varphi ^{(k)} (0)\over k!} x^k\right] dx +\sum _{k=0} ^{s-1} {\varphi ^{(k)} (0)\over k! (\lambda +k+1)} . \end{aligned}$$

The distribution $x_+^{-s}$ is defined by

$$\begin{aligned} x_+ ^{-s}= {(-1)^{s-1}( \ln x_+ )^{(s)} \over (s-1)!} \end{aligned}$$

for $s= 1,2, \ldots$ and not as in Gelfand and Shilov [5].

It is easily shown that if $\varphi$ is an arbitrary function in ${{\mathcal {D}}}[-1,1],$ then

$$\begin{aligned} \langle x_+^{-s} , \varphi (x) \rangle= & {} \int _0^{1} x^{-s} \left[ \varphi (x) - {\sum } _{k=0}^{s-1} \frac{\varphi ^{(k)}(0)}{k!} x^{k} \right] \,dx \nonumber \\&\,\,\, -{\sum }_{k=0}^{s-2} \frac{\varphi ^{(k)}(0)}{(s-k-1) k!} -{\phi (s-1)\over (s-1)!}\varphi ^{(s-1)}(0), \end{aligned}$$

(1)

for $s=1,2, \ldots ,$ where

$$\begin{aligned} \phi (s)=\left\{ \begin{array}{ll} \sum _{k=1} ^s k^{-1} , &{}s\ge 1,\\ 0, &{} s=0. \end{array} \right. \end{aligned}$$

The composition of a distribution and an infinitely differentiable function is extended to distributions by continuity provided that the derivative of the infinitely differentiable function is different from zero, see [1]. Fisher defined the composition of a distribution F and a summable function f which has a single simple root in the open interval (a, b) and it was recently generalized in [6] by allowing f to be a distribution.

Now let $\rho (x)$ be a function in ${{\mathcal {D}}}$ having the following properties:

• (i) $\rho (x) =0$ for $|x| \ge 1$,      (ii)   $\rho (x) \ge 0$,

• (iii) $\rho (x) =\rho (-x)$,      (iv)   ${\displaystyle {\int _{-1}^1 \rho (x)\,dx =1}}$.

Putting $\delta _n(x)=n\rho (nx)$ for $n=1,2,\ldots$ , it follows that $\{\delta _n(x)\}$ is a regular sequence of infinitely differentiable functions converging to the Dirac delta function $\delta (x)$.

Further, if F is a distribution in ${{\mathcal {D}}}'$ and $F_n(x)=\langle F(x-t),\delta _n(x)\rangle ,$ then $\{F_n(x)\}$ is a regular sequence of infinitely differentiable functions converging to F(x).

The following definition for the neutrix composition of distributions was given in [3] and originally called the composition of distributions.

Definition 1.1

Let F be a distribution in ${{\mathcal {D}}}'$ and let f be a locally summable function. We say that the neutrix composition F(f(x)) exists and is equal to h on the open interval (a, b) if

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }\int _{-\infty }^{\infty } F_{n}(f(x))\varphi (x) dx = \langle h(x) , \varphi (x) \rangle \end{aligned}$$

for all $\varphi$ in ${{\mathcal {D}}}[a,b],$ where $F_n(x) =F(x)*\delta _n(x)$ for $n=1,2, \ldots$ and N is the neutrix, see [2], having domain $N'$ the positive integers and range $N''$ the real numbers, with negligible functions which are finite linear sums of the functions

$$\begin{aligned} n^{\lambda } \ln ^{r-1} n, \, \ln ^{r} n: \lambda > 0, \, r=1,2,\ldots \end{aligned}$$

and all functions which converge to zero in the usual sense as n tends to infinity.

In particular, we say that the composition F(f(x)) exists and is equal to h on the open interval (a, b) if

$$\begin{aligned} \mathop {\mathrm{lim}}\limits _{n\rightarrow \infty }\int _{-\infty }^{\infty } F_{n}(f(x))\varphi (x) dx = \langle h(x) , \varphi (x) \rangle \end{aligned}$$

for all $\varphi$ in ${{\mathcal {D}}}[a,b].$

Note that if a function f(n) tends to $\alpha$ in the usual sense as n tends to infinity, it converges to $\alpha$ in the neutrix sense. The reader may find the general definition of the neutrix limit with some examples in [2].

2 Results

Nicholas and Fisher defined the composition $(x_+^r)^{-s}$ as the neutrix limit of regular sequence $[(x_+^r)^{-s}]_n$ for $r,s =1,2,\ldots ,$ see [7]. Further Öz c̣ ağ et. al. consider the case $r=0$, in other words the s-th power of the Heaviside function H(x) defined by $[H(x)]^{-s} =H(x),$ see [9]. Recently, some compositions such as $(|x|^{r-1/2})^{-4s}$ and $(|x|^\mu )^{-s}$ were defined in [4, 8] respectively.

We first of all need the following Lemma which can be easily proved by induction.

Lemma 2.1

$$\begin{aligned} \int _{-1} ^1 v^i \rho ^{(r)} (v) \,dv = \left\{ \begin{array} {cc} 0,&{} 0 \le i<r, \\ (-1)^r r!,&{} i=r \end{array} \right. \end{aligned}$$

(2)

and because $v^r\rho ^{(r)}$ is an even function, we have

$$\begin{aligned} \int _{-1}^0 v^r\rho ^{(r)}(v)\,dv =\int _0^1 v^r\rho ^{(r)}(v)\,dv = \frac{1}{2}(-1)^rr! \end{aligned}$$

(3)

and

$$\begin{aligned} \int _{-1}^0 v^r\ln |v|\rho ^{(r)}(v)\,dv= & {} \int _0^1 v^r\ln |v|\rho ^{(r)}(v)\,dv \nonumber \\= & {} \frac{1}{2}(-1)^rr!\phi (r)+(-1)^rr!c(\rho ) \end{aligned}$$

(4)

for $r=0,1,2,\ldots$ where $c(\rho )=\int _0^1\ln t\rho (t)\,dt.$

We now prove the following theorem.

Theorem 2.2

The distribution $(x_+^\mu )^{-m}$ exists and

$$\begin{aligned} (x_+^\mu )^{-m}=x_+^{-s}-(-1)^s\frac{(-1)^mm!\left[ 2c(\rho )+\phi (m-1)\right] +s\phi (s-1)}{s!}\delta ^{(s-1)}(x) \end{aligned}$$

(5)

for $\mu >0, \quad m=1,2,\ldots$ and $\mu m=s (s\in {{\mathbb {Z}}}^+).$

In particular, we have

$$\begin{aligned} \left( x_+^\frac{1}{r}\right) ^{-r} = x_+^{-1}-(-1)^rr!\left[ 2c(\rho )+\phi (r-1)\right] \delta (x) \end{aligned}$$

(6)

for $r=1,2,\ldots .$

Proof

We first put

$$\begin{aligned} (-1)^{m-1}(m-1)![(x_+^\mu )^{-m}]_n = \left\{ \begin{array}{cc} \int _{-1/n}^{1/n} \ln |x^\mu - t| \delta _n^{(m)}(t) dt , &{} if \quad x\ge 0 \\ \int _{-1/n}^{1/n} \ln |t| \delta _n^{(m)}(t) dt , &{} if \quad x<0. \end{array} \right. \end{aligned}$$

(7)

Then

$$\begin{aligned}&(-1)^{m-1}(m-1)! \int _{-1}^1 x^k [(x_+^\mu )^{-m}]_n\,dx = \int _{-1}^{1} x^k \int _{-1/n}^{1/n} \ln |x_+^\mu - t| \delta _n^{(m)}(t) dt dx \nonumber \\&\quad = \int _{-1/n}^{1/n} \delta _n ^{(m)} (t) \int _0^{n^{-1/\mu }} x^k \ln |x^\mu -t| \,dx \,dt \nonumber \\&\qquad +\int _{-1/n}^{1/n} \delta _n ^{(m)}(t) \int _{n^{-1/\mu }}^1 x^k \ln |x^\mu -t|\,dx \,dt +\int _{-1}^0 x^k \int _{-1/n}^{1/n} \ln |t|\delta _n ^{(m)}(t) \,dx\,dt \nonumber \\&\quad = \frac{n^{m-(k+1)/\mu }}{\mu } \int _{-1}^1 \rho ^{(m)}(v) \int _0 ^1 y^{-1+(k+1)/\mu }\ln |y-v| \,dy\,dv \nonumber \\&\qquad +\,\frac{n^{m-(k+1)/\mu }}{\mu } \int _{-1}^1 \rho ^{(m)}(v)\int _ 1^n y^{-1+(k+1)/\mu }\ln |y-v| \,dy\,dv \nonumber \\&\qquad +\,\frac{n^{m-(k+1)/\mu }}{\mu }\ln n\int _{-1}^1 \rho ^{(m)}(v)\,dv\int _0^n y^{-1+(k+1)/\mu }\,dy \nonumber \\&\qquad +\,\frac{(-1)^{k+1}n^m}{k+1}\int _{-1}^1\ln |v/n|\rho ^{(m)}(v)\,dv \nonumber \\&\quad = I_1 + I_2+I_3+I_4, \end{aligned}$$

(8)

on using the substitutions $y = nx^\mu$ and $v = nt.$

It follows immediately that

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }I_3 = 0\qquad and \qquad \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }I_4 = 0 \end{aligned}$$

(9)

for $k=0,1,\ldots$ and

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }I_1 = 0 \end{aligned}$$

(10)

for all $k=0,1,2,\ldots ,s-2.$

Further

$$\begin{aligned} \int _1^n y^{-1+(k+1)/\mu }\ln |y-v| \,dy= & {} \int _1^n y^{-1+(k+1)/\mu }\ln y\,dy \nonumber \\&+\int _1^n y^{-1+(k+1)/\mu }\ln |1-v/y| \,dy \nonumber \\= & {} I_2' +I_2'', \end{aligned}$$

(11)

where

$$\begin{aligned} I_2' =\frac{\mu n^{(k+1)/\mu } \ln n}{(k+1)}+\frac{\mu ^2[1-n^{(k+1)/\mu } ]}{(k+1)^2} \end{aligned}$$

(12)

and

$$\begin{aligned} I_2''= & {} -\sum _{i=1}^\infty \frac{v^i}{i}\int _1^n y^{-1-i+(k+1)\mu } \,dy \nonumber \\= & {} -\sum _{i=1}^\infty \frac{v^i\mu [n^{-i+(k+1)/\mu }-1]}{i(k+1-\mu i)} \end{aligned}$$

(13)

$k=0,1,2,\ldots ,s-2.$

It follows from Lemma 2.1 and Eqs. (11), (12), and (13) that

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }I_2 =\frac{(-1)^m (m-1)!}{s-k-1} \end{aligned}$$

(14)

for $k=0,1,\ldots ,s-2,$ and it then follows from Eqs. (8), (9), (10), and (14) that

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }\int _{-1}^1 x^k \left[ (x_+^\mu )^{-m} \right] _n \,dx = -\frac{1}{s-k-1} \end{aligned}$$

(15)

$k=0,1,2,\ldots ,s-2.$

When $k=s-1,$ then we have

$$\begin{aligned} I_1= & {} \frac{1}{\mu }\int _{-1}^1\rho ^{(m)}(v) \int _0^1 y^{m-1}\ln |y-v| \,dy\,dv \nonumber \\= & {} \frac{1}{\mu } \int _0^1\rho ^{(m)}(v) \left[ \int _0^v + \int _v^1 y^{m-1} \ln |y-v|\,dy\right] \,dv \nonumber \\&\,\,\, + \frac{1}{\mu } \int _{-1}^0 \rho ^{(m)}(v) \left[ \int _0^{-v} + \int _{-v}^1 y^{m-1} \ln |y-v| \,dy\right] \,dv \nonumber \\= & {} J_1+J_2+J_3+J_4. \end{aligned}$$

(16)

Now making the substitution $y=uv$

$$\begin{aligned} J_1 = \frac{1}{\mu } \int _0^1 v^m \rho ^{(m)}(v)\int _0^1 u^{m-1}\left[ \ln v+\ln (1-u)\right] \,du\,dv, \end{aligned}$$

(17)

and using Lemma 2.1, we have

$$\begin{aligned} \int _0^1 v^m \rho ^{(m)}(v)\ln v\int _0^1 u^{m-1}\,du\,dv =(-1)^m(m-1)! \left[ c(\rho ) +\frac{1}{2}\phi (m) \right] \end{aligned}$$

(18)

and

$$\begin{aligned}&\int _0^1 v^m \rho ^{(m)}(v)\int _0^1 u^{m-1}\ln (1-u)\,du\,dv \nonumber \\&\quad =\frac{1}{2}(-1)^m(m-1)!\int _0^1\ln (1-u)d(u^m-1)\nonumber \\&\quad = \frac{1}{2}(-1)^m(m-1)!\int _0^1\frac{u^m-1}{1-u}\,du \nonumber \\&\quad = \frac{1}{2}(-1)^{m-1}(m-1)!\phi (m), \end{aligned}$$

(19)

and it follows from Eqs. (17), (18), and (19) that

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }J_1 = \frac{(-1)^m(m-1)! c(\rho )}{\mu }. \end{aligned}$$

(20)

Similarly, using the substitution $y=uv$ again we have

$$\begin{aligned} J_3 = - \frac{1}{\mu } \int _{-1}^0 v^m \rho ^{(m)}(v)\int _{-1}^0 u^{m-1}\left[ \ln |v|+\ln (1-u)\right] \,du\,dv. \end{aligned}$$

(21)

Thus

$$\begin{aligned} \int _{-1}^0 v^m \rho ^{(m)}(v)\ln |v|\int _{-1}^0 u^{m-1}\,du\,dv= & {} \frac{(-1)^{m-1}}{m}\int _{-1}^0 v^m \rho ^{(m)}(v)\ln |v|\,dv \nonumber \\= & {} -(m-1)!\left[ c(\rho )+\frac{1}{2}\phi (m) \right] , \end{aligned}$$

(22)

and

$$\begin{aligned}&\int _{-1}^0 v^m \rho ^{(m)}(v)\int _{-1}^0 u^{m-1}\ln (1-u)\,du\,dv \nonumber \\&\quad =\frac{1}{2}(-1)^m(m-1)!\int _{-1}^0\ln (1-u)d(u^m-1) \nonumber \\&\quad =\frac{1}{2}[(-1)^m-1](m-1)!\ln 2-\frac{1}{2}(-1)^m(m-1)!\int _{-1}^0\frac{u^m-1}{u-1}\,du \nonumber \\&\quad = \frac{1}{2}[(-1)^m-1](m-1)!\ln 2+\frac{1}{2}(-1)^m(m-1)!\sum _{i=1}^m \frac{(-1)^i}{i}. \end{aligned}$$

(23)

It follows from Eqs. (21), (22), and (23) that

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }J_3= & {} \frac{[1-(-1)^m] (m-1)!}{2\mu }\ln 2+\frac{(m-1)!}{2\mu }\phi (m)+\nonumber \\&\,\,\,-\frac{(m-1)!}{2\mu }\sum _{i=1}^m \frac{(-1)^i}{i}+\frac{(m-1)!}{\mu }c(\rho ). \end{aligned}$$

(24)

Further

$$\begin{aligned} J_2= & {} \frac{1}{\mu }\int _0^1 \rho ^{(m)}(v)\int _v^1 y^{m-1} \left[ \ln y+\ln (1-v/y)\right] \,dy\,dv \nonumber \\= & {} \frac{1}{\mu }\int _0^1 \rho ^{(m)}(v)\int _v^1 y^{m-1}\ln y\,dy\,dv-\frac{1}{\mu }\sum _{i=1}^\infty \frac{1}{i}\int _0^1 v^i\rho ^{(m)}(v)\int _v^1 y^{m-i-1}\,dy\,dv \nonumber \\= & {} \frac{(-1)^m(m-1)!}{2\mu m}-\frac{1}{\mu m^2}\int _0^1 \rho ^{(m)}(v)\,dv + \nonumber \\&\,\,\,-\frac{1}{\mu }\sum _{i=1, \,i\ne m}^\infty \frac{1}{i(m-i)}\int _0^1(v^i-v^m)\rho ^{(m)}(v)\,dv \nonumber \\= & {} \frac{(-1)^m(m-1)!}{2\mu m}+\frac{\rho ^{(m-1)}(0)}{\mu m^2}+\frac{(-1)^m(m-1)!}{2\mu } \left[ 2\phi (m-1)- \phi (m)\right] \nonumber \\&\,\,\,-\frac{1}{\mu }\sum _{i=1, \,i\ne m}^\infty \frac{1}{i(m-i)}\int _0^1v^i\rho ^{(m)}(v)\,dv \nonumber \\= & {} \frac{\rho ^{(m-1)}(0)}{\mu m^2}+\frac{(-1)^m(m-1)!}{2\mu }\phi (m-1)+ \nonumber \\&\,\,\, -\frac{1}{\mu }\sum _{i=1, \,i\ne m}^\infty \frac{1}{i(m-i)}\int _0^1v^i\rho ^{(m)}(v)\,dv \end{aligned}$$

(25)

because

$$\begin{aligned} \sum _{i=1, \,i\ne m}^\infty \frac{1}{i(m-i)} =\frac{2\phi (m-1)-\phi (m)}{m}= \frac{\phi (m-1)}{m}-\frac{1}{m^2}. \end{aligned}$$

Finally, we have

$$\begin{aligned} J_4= & {} \frac{1}{\mu }\int _{-1}^0 \rho ^{(m)}(v)\int _{-v}^1 y^{m-1} \left[ \ln y+\ln (1-v/y)\right] \,dy\,dv \nonumber \\= & {} \frac{1}{\mu }\int _{-1}^0 \rho ^{(m)}(v)\int _{-v}^1 y^{m-1}\ln y\,dy\,dv-\frac{1}{\mu }\sum _{i=1}^\infty \frac{1}{i}\int _{-1}^0 v^i\rho ^{(m)}(v)\int _{-v}^1 y^{m-i-1}\,dy\,dv \nonumber \\= & {} \frac{1}{\mu }\int _{-1}^0\left[ \frac{(-v)^m-1}{m^2}-\frac{(-v)^m\ln |v|}{m} \right] \rho ^{m)}(v)\,dv+ \frac{1}{\mu m}\int _{-1}^0 v^m\ln |v|\rho ^{(m)}(v)\,dv \nonumber \\&\,\,\,-\frac{1}{\mu }\sum _{i=1, \,i\ne m}^\infty \frac{1}{i(m-i)}\int _{-1}^0 \left[ v^i-(-1)^{m-i}v^m) \right] \rho ^{(m)}(v)\,dv \nonumber \\= & {} \frac{(m-1)!}{2\mu m}-\frac{\rho ^{(m-1)}(0)}{\mu m^2}-\frac{[1-(-1)^m]m!}{2\mu } \left[ \phi (m)+2c(\rho )\right] + \nonumber \\&\,\,\,-\frac{1}{\mu }\sum _{i=1, \,i\ne m}^\infty \frac{1}{i(m-i)}\int _{-1}^0 v^i \rho ^{(m)}(v)\,dv-\frac{(-1)^m(m-1)!}{2\mu m} \nonumber \\&\,\,\,+\frac{(m-1)!}{2\mu }\sum _{i=1}^{m-1} \frac{(-1)^{m+i}}{i} -\frac{[1-(-1)^m] (m-1)!\ln 2}{2\mu } \end{aligned}$$

(26)

because

$$\begin{aligned} \sum _{i=1, \,i\ne m}^\infty \frac{1}{i(m-i)} =-\frac{(-1)^m}{m^2}+\frac{1}{m}\sum _{i=1}^{m-1} \frac{(-1)^{m+i}}{i}- \frac{[1-(-1)^m]}{m}\ln 2 . \end{aligned}$$

Next, we similarly have

$$\begin{aligned} I_2= & {} \frac{1}{\mu }\int _{-1}^1 \rho ^{(m)}(v)\int _1^n y^{m-1} \left[ \ln y+\ln (1-v/y)\right] \,dy\,dv \\= & {} \frac{1}{\mu }\int _{-1}^1 \rho ^{(m)}(v)\int _1^n y^{m-1} \ln (1-v/y) \,dy\,dv \\= & {} -\frac{1}{\mu }\sum _{i=1}^\infty \frac{1}{i} \int _{-1}^1 v^i\rho ^{(m)}(v) \int _1^n y^{m-i-1}\,dy\,dv \\= & {} -\frac{(-1)^m(m-1)!\ln n}{\mu }-\frac{1}{\mu }\sum _{i=m+1}^\infty \frac{1}{i(i-m)} \int _{-1}^1(n^{m-i}-1)v^i\rho ^{(m)}(v)\,dv, \end{aligned}$$

and so

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }I_2 = -\frac{1}{\mu }\sum _{i=m+1}^\infty \frac{1}{i(i-m)} \int _{-1}^1 v^i\rho ^{(m)}(v)\,dv. \end{aligned}$$

(27)

It now follows from Eqs. (8), (9), (16), (20), and (24)–(27) that

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }\int _{-1}^1 x^{s-1}\left[ (x_+^\mu )^{-m} \right] _n \,dx= & {} \frac{(-1)^m(m-1)!}{\mu } \left[ 2c(\rho )+\phi (m-1) \right] \nonumber \\= & {} \frac{(-1)^mm!}{s} \left[ 2c(\rho )+\phi (m-1) \right] . \end{aligned}$$

(28)

We now consider the case $k=s.$ If $x<0$ and $\psi$ is an arbitrary continuous function, then

$$\begin{aligned}&(-1)^{m-1}(m-1)!\int _{-1}^0 x^s \left[ (x_+^\mu )^{-m} \right] _n \psi (x)\,dx \\&\quad =\int _{-1}^0 x^s \psi (x)\int _{-1/n}^{1/n} \ln |t| \delta _n^{(m)}(t)\,dt\,dx \\&\quad =n^m \int _{-1}^0 x^s\psi (x)\,dx \int _{-1}^1 \ln |v/n|\rho ^{(m)}(v)\,dv, \end{aligned}$$

where $v=nt.$ Thus

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }\int _{-1}^0 x^s \left[ (x_+^\mu )^{-m}\right] _n \psi (x)\,dx =0. \end{aligned}$$

(29)

Next with $k=s$ in $I_1,$ we see that

$$\begin{aligned} \int _0^{n^{-1/\mu }} x^s \left[ (x_+^\mu )^{-m} \right] _n \,dx=\frac{n^{-1/\mu }}{\mu } \int _{-1}^1 \rho ^{(m)}(v)\int _0^1 y^{m-1+1/\mu } \ln |(y-v)/n| \,dy\,dv \end{aligned}$$

and if $\psi$ is any continuous function, then

$$\begin{aligned} \mathop {\mathrm{lim}}\limits _{n\rightarrow \infty }\int _0^{n^{-1/\mu }} x^s \left[ (x_+^\mu )^{-m} \right] _n \psi (x)\,dx =0. \end{aligned}$$

(30)

If $x^\mu \ge \frac{1}{n}$, then we have

$$\begin{aligned} (-1)^{m-1}(m-1)!\left[ (x_+^\mu )^{-m} \right] _n= & {} \int _{-1/n}^{1/n} \ln |x^\mu -t| \delta _n^{(m)}(t)\,dt \\= & {} n^m \int _{-1}^1 \ln |x^\mu -v/n| \rho ^{(m)}(v)\,dv \\= & {} n^m \int _{-1}^1\left[ \ln |x^\mu -\sum _{i=1}^\infty \frac{v^i}{in^ix^{\mu i}} \right] \rho ^{(m)}(v)\,dv \\= & {} -\sum _{i=m}^\infty \int _{-1}^1\frac{v^i}{in^{i-m}x^{\mu i}}\rho ^{(m)}(v)\,dv, \end{aligned}$$

and it follows that

$$\begin{aligned} \left| (m-1)!\left[ (x_+^\mu )^{-m} \right] _n \right|\le & {} \sum _{i=m}^\infty \int _{-1}^1\frac{|v|^i}{in^{i-m}x^{\mu i}}|\rho ^{(m)}(v)|\,dv \\\le & {} \sum _{i=m}^\infty \frac{\kappa _m}{in^{i-m}x^{\mu i}}, \end{aligned}$$

where $\kappa _m =\int _{-1}^1 |\rho ^{(m)}(v)|\,dv$ for $m=1,2,\ldots .$

If now $n^{-1/\mu }<\eta <1$

$$\begin{aligned}&(m-1)! \int _{n^{-1/\mu }}^\eta \left| x^s \left[ (x_+^\mu )^{-m} \right] _n \right| \,dx \le \kappa _m\sum _{i=m}^\infty \frac{n^{m-i}}{i} \int _{n^{-1/\mu }}^\eta x^{s-\mu i} \,dx \\&\quad =\kappa _m \sum _{i=m}^\infty \frac{n^{-1/\mu }}{\mu i} \int _1^{n\eta ^\mu } y^{m-i+1/\mu -1}\,dy \\&\quad =\left\{ \begin{array}{ll} \kappa _m \sum _{i=m}^\infty \frac{n^{-1/\mu }}{\mu i(m-i+1/\mu )} \left[ (n\eta ^\mu )^{m-i+1/\mu }-1\right] , &{} \mu \ne 1 \\ \kappa _m \sum _{i=m,i\ne m+1}^\infty \frac{n^{-1}}{i(m-i+1)} \left[ (n\eta )^{m-i+1}-1\right] +\frac{\kappa _m n^{-1}\ln (n\eta )}{m+1} , &{} \mu =1.\end{array} \right. \end{aligned}$$

It follows that

$$\begin{aligned} \mathop {\mathrm{lim}}\limits _{n\rightarrow \infty }\left| \left[ (x_+^\mu )^{-m} \right] _n \right| = O(\eta ) \end{aligned}$$

for $m=1,2,\ldots$ and if $\psi$ is any continuous function then

$$\begin{aligned} \mathop {\mathrm{lim}}\limits _{n\rightarrow \infty }\left| \int _{n^{-1/\mu }}^\eta x^s \left[ (x_+^\mu )^{-m} \right] _n \psi (x)\,dx \right| = O(\eta ) \end{aligned}$$

(31)

for $m=1,2,\ldots .$ Now let $\varphi (x)\in {{\mathcal {D}}}[-1,1].$ By Taylor’s theorem, we have

$$\begin{aligned} \varphi (x) = \sum _{k=0}^{s-1}\frac{\varphi ^{(k)}(0)}{k!} x^k +\frac{x^s}{s!} \varphi ^{(s)}(\xi x) \qquad (0<\xi <1). \end{aligned}$$

Then

$$\begin{aligned} \langle \left[ (x_+^\mu )^{-m} \right] _n,\varphi (x) \rangle= & {} \sum _{k=0}^{s-1}\frac{\varphi ^{(k)}(0)}{k!}\int _{-1}^1 x^k \left[ (x_+^\mu )^{-m} \right] _n \,dx \\&+\frac{1}{s!}\int _{-1}^0 x^s\left[ (x_+^\mu )^{-m} \right] _n \varphi ^{(s)}(\xi x)\,dx \\&+\frac{1}{s!} \int _0^{n^{-1/\mu }} x^s \left[ (x_+^\mu )^{-m} \right] _n\varphi ^{(s)}(\xi x)\,dx \\&+\frac{1}{s!}\int _{n^{-1/\mu }}^\eta x^s \left[ (x_+^\mu )^{-m} \right] _n\varphi ^{(s)}(\xi x)\,dx \\&+\frac{1}{s!} \int _\eta ^1 x^s \left[ (x_+^\mu )^{-m} \right] _n\varphi ^{(s)}(\xi x)\,dx. \end{aligned}$$

Using Eqs. (15) and (28)–(31) and noting that the sequence $\{ \left[ (x_+^\mu )^{-m} \right] _n \}$ converges uniformly to $x^{-s}$ on the interval $[\eta ,1]$, it follows that

$$\begin{aligned} \mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }\langle \left[ (x_+^\mu )^{-m} \right] _n,\varphi (x) \rangle= & {} \frac{(-1)^mm!}{s!} \left[ 2c(\rho )+\phi (m-1)\right] \varphi ^{(s-1)}(0) \\&\,\,\, -\sum _{k=0}^{s-2}\frac{\varphi ^{(k)}(0)}{k!(s-k-1)} +\int _\eta ^1 \varphi ^{(s)}(\xi x) \,dx+ O(\eta ). \end{aligned}$$

Because $\eta$ can be arbitrarily small, it follows that

$$\begin{aligned}&\mathop {\mathrm{N}-\mathrm{lim}}\limits _{n\rightarrow \infty }\langle \left[ (x_+^\mu )^{-m} \right] _n,\varphi (x) \rangle \nonumber \\&\quad =\frac{(-1)^mm!}{s!} \left[ 2c(\rho )+\phi (m-1)\right] \varphi ^{(s-1)}(0)-\sum _{k=0}^{s-2}\frac{\varphi ^{(k)}(0)}{k!(s-k-1)}\\&\qquad +\int _\eta ^1 \varphi ^{(s)}(\xi x) \,dx+ O(\eta ). \\&\quad =\int _0^1 x^{-s} \left[ \varphi (x)-\sum _{k=0}^{s-1} \frac{\varphi ^{(k)}(0)}{k!} x^k \right] \,dx-\sum _{k=0}^{s-2} \frac{\varphi ^{(k)}(0)}{k!(s-k-1)} \\&\qquad +\frac{(-1)^mm!}{s!} \left[ 2c(\rho )+\phi (m-1)\right] \varphi ^{(s-1)}(0) \\&\quad =\langle x_+^{-s}, \varphi (x) \rangle + \\&\qquad -(-1)^s\frac{(-1)^mm!\left[ 2c(\rho )+\phi (m-1)\right] +s\phi (s-1)}{s!}\langle \delta ^{(s-1)}(x),\varphi (x)\rangle \end{aligned}$$

on using Eq. (1). This proves Eq. (5) on the interval $[-1,1].$ However, Eq. (5) clearly holds on any interval not containing the origin, and the proof is complete. $\square$

Note that if $\mu \in {{\mathbb {Z}}}^+,$ then Theorem 1 is in agreement with the Theorem 3 given in [7].

Corollary 2.3

The distribution $(x_-^\mu )^{-m}$ exists and

$$\begin{aligned} (x_-^\mu )^{-m}=x_-^{-s}+\frac{(-1)^mm!\left[ 2c(\rho )+\phi (m-1)\right] +s\phi (s-1)}{s!}\delta ^{(s-1)}(x) \end{aligned}$$

(32)

for $\mu >0, \quad m=1,2,\ldots$ and $\mu m=s\in {{\mathbb {Z}}}^+.$

Proof

Equation (32) follows after replacing x by $-x$ in Eq. (5). $\square$