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The composition of the distributions \(x_-^\lambda \ln^s x_-\) and \(x_+^{- r / \lambda}\). (English) Zbl 1455.46045

Summary: The composition \(F(f(x))\) of a distribution \(F(x)\) and a locally summable function \(f(x)\) is defined as the neutrix limit of the regular sequence \(\{ F_n(f(x))\}\). In this paper, we prove that the neutrix composition of the distributions \(x_-^\lambda \ln^s x_-\) and \(x_+^{- r / \lambda}\) exists and equals \[\frac{ ( -1 )^r \lambda}{ r !} \sum_{i = 0}^s \sum_{j = 0}^i\binom{i}{j} \binom{s}{i}\frac{ c_{m , i} B_{s - i , 0} ( \lambda + 1 , m ) B_{0 , i - j} ( - \lambda , \lambda + m + 1 )}{ ( m - 1 ) !} \delta^{( r - 1 )}(x),\] for \(\lambda<0\), \(\lambda\neq-1,-2,\dots\), \(r=1,2,\dots\) and \(s=0,1,2,\dots\), where \(B(\lambda,\mu)\) denotes the Beta function, \[ B_{i , j}(\lambda,\mu)=\frac{ \partial^{i + j}}{ \partial^i \lambda \partial^j \mu}B(\lambda,\mu),\] for \(i,j=0,1,2,\dots\), and \[c_{m , i}= \int_0^1 v^m \ln^iv \rho^{( m )}(v)\,dv,\] for \(i=0,1,2,\dots,s\) and \(-m-1<\lambda<-m\), for \(m=1,2,\dots\).

MSC:

46F10 Operations with distributions and generalized functions
41A30 Approximation by other special function classes
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