鍏抽敭璇�: 瀵硅瀹氱悊, D-鏈夐檺, P-閫掑綊

Abstract: Special functions that satisfy linear differential equations with polynomial coefficients appear ubiquitously in combinatorics and mathematical physics. Such kind of special functions are called D-finite functions by Stanley. In the early 1980's, many combinatorists, such as Gessel, Stanley, Zeilberger etc., conjectured that the diagonal of rational power series in several variables is D-finite. Gessel and Zeilberger proved this conjecture in their papers, respectively. Later, Lipshitz pointed out that their proofs are not complete and he gave a proof by basing on a different idea. Zeilberger completed his proof with the theory of holonomic D-modules. In this note, we follow the spirit of Gessel's proof strategy and fix the gap in his proof in the case of bivariate rational formal power series. The key ingredients we used are some basic properties of the diagonal operation.

Key words: Diagonal theorem, D-finite, P-recursive