Residue methods in combinatorial analysis

This article studies the use of residues defined on local cohomology classes as a computing tool for combinatorial analysis. Such cohomology residues are used globally or locally to prove binomial identities. Invariance of residues under variable change gives rise to more identities and explains phenomena such as Lagrange inversion formula and Riordan array. It also leads to the discovery of Schauder bases, which are used to characterize inverse relations with orthogonal property. Furthermore residues are analyzed to compute the diagonal of a rational power series. Many classical numbers can be represented by residues. New insights of certain numbers are discovered via computations on these representations. Finally, we extend ground field so that polynomial sequences can be treated as scalars sequences. The methods of cohomology residues for scalars thus carries over.