Recent Results for the Q-lagrange Inversion Formula
نویسندگان
چکیده
A survey of the q-Lagrange inversion formula is given, including recent work of Garsia, Gessel, Hofbauer, Krattenthaler, Remmel, and Stanton. Some applications to identities of Rogers-Ramanujan type are stated.
منابع مشابه
q-Catalan Numbers
q-analogs of the Catalan numbers c', = (I/(n + I))($) are studied from the viewpoint of Lagrange inversion. The first, due to Carhtz, corresponds to the Andrews-Gessel-Garsia q-Lagrange inversion theory, satisfies a nice recurrence relation and counts inversions of Catalan words. The second, tracing back to Mac Mahon, arise from Krattenthaler's and Gessel and Stanton's q-Lagrange inversion form...
متن کاملLagrange Inversion and Schur Functions
Macdonald defined an involution on symmetric functions by considering the Lagrange inverse of the generating function of the complete homogeneous symmetric functions. The main result we prove in this note is that the images of skew Schur functions under this involution are either Schur positive or Schur negative symmetric functions. The proof relies on the combinatorics of Lagrange inversion. W...
متن کاملA One-parameter Deformation of the Noncommutative Lagrange Inversion Formula
We give a one-parameter deformation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder-Frabetti-Krattenthaler for the antipode of the noncommutative Faá di Bruno algebra. Namely, we obtain a closed formula for the antipode of the one-parameter deformation of this Hopf algebra discovered by Foissy.
متن کاملA Physicist’s Proof of the Lagrange-Good Multivariable Inversion Formula
We provide yet another proof of the classical Lagrange-Good multivariable inversion formula using the techniques of quantum field theory.
متن کاملLagrange Inversion for Species
1. Introduction. The Lagrange inversion formula is one of the fundamental results of enumerative combinatorics. It expresses the coefficients of powers of the compositional inverse of a power series in terms of the coefficients of powers of the original power series. G. Labelle [10] extended Lagrange inversion to cycle index series, which are equivalent to symmetric functions. Although motivate...
متن کامل