On the kernel of sequence transformations
The paper deals with the production of constant sequences by use of a transformation based upon difference sequences. Let \({\mathcal G}\) be a system of sequences \({\mathcal G} [i] (i \geq 0)\), the members of \({\mathcal G} [i]\) being \(G(i | n)\) \((n \geq 0)\), \({\mathcal P}\) and \({\mathcal Q}\) be two sequences of integers \(p(n) \geq 0\) \((n \geq 0)\) and \(q(n)\) \((n \geq 0)\) and \({\mathcal F}\) be the sequence whose terms are \(F(n)\) \((n \geq 0)\). The members of the difference sequence \(\delta ({\mathcal F}, {\mathcal G} : {\mathcal P}, {\mathcal Q})\) of \({\mathcal F}\) are \(\delta ({\mathcal F}, {\mathcal G} : {\mathcal P}, {\mathcal Q} | n) = \sum F(i + n) G(i | n)\) \((- p(n) \leq i \leq q(n))\), being defined for \(n \geq 0\), where \(F(i + n) = 0\) when \(i + n < 0\). \(({\mathcal G}, {\mathcal P}\) and \({\mathcal Q}\) are fixed throughout: \(\delta ({\mathcal F}, {\mathcal G} : {\mathcal P}, {\mathcal Q})\) is written as \(\delta ({\mathcal F}).)\) \(L\) being constant, a relationship of the form \(\delta (L{\mathcal F}) = L \delta ({\mathcal F})\) evidently holds. \({\mathcal S}\) and \({\mathcal E}\) being sequences whose members are \(S(n)\) and \(E(n)\) respectively \((n \geq 0)\), the sequence \({\mathcal S} {\mathcal E}\) is defined by its members \(S(n) E(n)\) \((n \geq 0)\). Assuming no member of \(\delta ({\mathcal E})\) to be zero, the members of \({\mathcal T} ({\mathcal S} : {\mathcal E})\), the transform of \({\mathcal S}\) by \({\mathcal E}\), are \({\mathcal T} ({\mathcal S} : {\mathcal E} | n) = \delta ({\mathcal S} {\mathcal E} | n)/ \delta ({\mathcal E} | n)\) \((n \geq 0)\). \({\mathcal A}\) being the sequence whose members are \(\{L - S(n)\} E(n)\) \((n \geq 0)\), it is shown that if \(\delta ({\mathcal A} | n) = 0\) for \(n \geq N\), then \({\mathcal T} ({\mathcal S} : {\mathcal E} | n) = L\) for \(n \geq N\). Further results of this sort related to recursively transformed sequences are stated and illustrated.
scientific article; zbMATH DE number 721004