Using basic tools from commutative algebra, a new and elementary framework is constructed to
host algebraic operations, which are also provided by analytic or cohomological residues. Lagrange inversion formulas are built into the framework. Local duality is proved.
Key Words: Differential, generating function, Lagrange inversion, power series ring, residue.
2010 Mathematics Subject Classification: Primary 13F25; Secondary 32A27.
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