Interpolation of rational functions by simple partial fractions (Q1762526)

The author studies the problem of the interpolation of a rational function \(F:\mathbb{C}\rightarrow\mathbb{C}\) by simple partial fractions of degree \(n\), i.e., of the form \(R:=Q^{\prime}/Q\), \(Q(x):=x^{n}+q_{n-1} x^{n-1}+\dotsb+q_{0}\), \(x\), \(q_{n}\in\mathbb{C}\). The problem is to find coefficients \(q_{j}\), \(j=\overline{0,n-1}\), such that \(R(z_{k})=y_{k} =f(z_{k})\), \(k=\overline{1,n}\), where \(z_{1},z_{2},\dotsc,z_{n}\in\mathbb{C}\) is a system of nodes such that \(Q(z_{k})\neq0\), \(k=\overline{1,n}\). This problem is reduced to the study of the solvability for a special difference equation. The author finds the subclass of rational functions for which an interpolating simple partial fractions of degree \(n\) is defined by a difference equation of order 1, and constructs explicit interpolation formulas for the functions \(1/(Ax+B)\), \(A\neq0\); \(1/(Ax^{2}+Bx)\), \(A\neq0\); \((Ax+B)/x\), \((AB)\neq0\); \((Ax+B)/(x^{2}+Cx)\), \((AB)\neq0\). Finally, he considers the interpolation by simple partial fractions of degree \(n\) of the functions \(A(x-a)^{m}\), \(m\in\mathbb{Z}\).