On the R. A. Sack theorem for translation operators
In this paper, I suggest an algorithm allowing one to find the ordered form of a polynomial (or of a power series) of the variables \(A\) and \(B\) satisfying the commutation relation \[ [A,B]= AB- BA=aB, \] where \(a\) is a given constant (or a variable commuting with \(A\) and \(B)\). This algorithm, whose statement is given in Section 4 of this paper, generalizes \textit{R. A. Sack}'s theorem [Philos. Mag., VIII. Ser. 3, 497-503 (1958; Zbl 0081.11701), formula (2.11)] asserting that for any polynomial \(F(\cdot)\) (or, at least formally, in a more general situation) the equalities \[ \begin{aligned} F(A+B) & =\exp \bigl\{(B_1/a) \Delta_a \bigr\}\bigl[ F(A) \bigr],\\ F(A+B) & =\exp \bigl\{(B_2/a) \Delta_a\bigr\} \bigl[F(A)\bigr] \end{aligned} \] are valid. Here, \(\Delta_a\) is the finite-difference operator acting on \(F(A)\): \[ \Delta_a \bigl[F(A) \bigr]= F(A+a) -F(A), \] and the subscript in \(B_1\) (respectively, in \(B_2)\) indicates that, when necessary operations are performed in every monomial, the factor containing \(B\) is placed to the left (respectively, to the right) of the factor containing \(A\). The algorithm, stated as Theorems 3 and 4 in Section 4, allows one to find the ordered form \(F(C)\), when the argument \(C\) has the form \(\varphi (B)A+ \psi(B)\) for any functions \(\varphi (B)\) and \(\psi(B)\).
scientific article; zbMATH DE number 1050683