On generalized Q-systems

On some Generalized q-Eulerian Polynomials

The (q, r)-Eulerian polynomials are the (maj−exc, fix, exc) enumerative polynomials of permutations. Using Shareshian and Wachs’ exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical q-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han’s three-variable statistic (inv−lec, pix, lec). We also prove a new ...

On Q-analogies of generalized Hermite’s polynomials

In this paper, we discuss the properties of q polynomials which are hold in any way, especially difference-differential equation and similar relations. Also, the distribution of zeros is studied. At last, we illustrate all by a few examples and make some conjectures.

A Note on Generalized q-Boole Polynomials

Let p be a prime number with p ≡ 1(mod 2). Throughout this paper, Zp,Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp. The p-adic norm | · |p is normalized as |p|p = 1 p . Let q be an indeterminate in Cp such that |1− q|p < p −1 p−1 . The q-extension of number x be defined as [x]q = 1−qx 1−q . Note that limq→1[...

PRESENTING GENERALIZED q - SCHUR

Generalized Q-functions

The modulus squared of a class of wave functions defined on phase space is used to define a generalized family of Q or Husimi functions. A parameter λ specifies orderings in a mapping from the operator |ψ〉〈σ| to the corresponding phase space wave function, where σ is a given fiducial vector. The choice λ = 0 specifies the Weyl mapping and the Q-function so obtained is the usual one when |σ〉 is ...