在组合数学与数学物理中,许多特殊函数满足系数为多项式的线性微分方程. 这类函数被称为D-有限函数. 上世纪80年代, Gessel, Stanley, Zeilberger等组合学家猜想多变元有理形式幂级数的对角是D-有限的. Gessel和Zeilberger分别在其文章中给出了该猜想的证明. 但是, Lipshitz在其文章中指出他们的证明是不完备的.本文基于对角算子的一些基本性质, 给出了两个变元情形下Gessel证明的更直接的修补办法.

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.