Multiple Derivative Inversions and Lagrange-Good Expansion Formulae

Abstract

By establishing new multiple inverse series relations (with their connection coefficients being given by higher derivatives of fixed multivariate analytic functions), we illustrate a general framework to provide new proofs for MacMahon’s master theorem and the multivariate expansion formula due to Good (1960). Further multivariate extensions of the derivative identities due to Pfaff (1795) and Cauchy (1826) will be derived and the generalized multifold convolution identities due to Carlitz (1977) will be reviewed.

Keywords

• leibniz rule
• Pfaff and Cauchy derivative identities
• inverse series relations
• MacMahon’s master theorem
• good’s formula of multivariate Lagrange expansion