A Hopf algebra structure on rational functions
Linearly recursive sequences over a field \(k\) appear as the continuous dual \(P^0\) of the Hopf algebra \(P = k[x]\) of polynomials with \(x\) primitive, e.g., see the paper of \textit{B. Peterson} and the reviewer [Aequationes Math. 20, 1-17 (1980; Zbl 0434.16008)]. \(P^0\) is a Hopf algebra with divided power product. As a vector space, \(P^0\) can be identified with rational functions \(p(x)/q(x)\), where \(q(0)\neq 0\), via the usual power-series expansion. In the paper under review, the author derives the same Hopf algebra structure when \(k\) is the field of complex numbers, but by a different procedure. He defines an inner product on the space \(Q\) of rational functions, and considers its restriction to \(P\times R\), where \(R\) is the space of rational functions \(p(x)/q(x)\), where \(\text{degree }p(x)<\text{degree }q(x)\). Then \(R\) embeds in the full linear dual of \(P\), and the Hopf algebra structure on \(P\) yields a Hopf algebra structure on \(R\). The inner product is then used to identify \(R\) with the space of linearly recursive sequences. There is also a discussion of the case when \(x\) is group-like, which yields the Hadamard product on \(R\).
scientific article; zbMATH DE number 863623