English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The classical Eulerian polynomials can be expanded in the basis$t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positiveintegral coefficients. This formula implies both the symmetry and theunimodality of the Eulerian polynomials. In this paper, we prove a $q$-analogueof this expansion for Carlitz's $q$-Eulerian polynomials as well as a similarformula for Chow-Gessel's $q$-Eulerian polynomials of type $B$. We shall givesome applications of these two formulae, which involve two new sequences ofpolynomials in the variable $q$ with positive integral coefficients. An openproblem is to give a combinatorial interpretation for these polynomials.

Two new triangles of $q$-integers via $q$-Eulerian polynomials of type $A$ and $B$