We notice that dual Grothendieck polynomials are specializations of some vexillary Schubert polynomials. Hence they have determinantal expressions in terms of complete or elementary symmetric functions, as well as a description in terms of tableaux and Giambelli type formula. We give for them a finite sum Cauchy identity.

The Ramanujan $_{1} \psi_{1}$ summation theorem is studied from the perspective of Jackson integrals, $q$-difference equations and connection formulae. This is an approach which has previously been shown to yield Bailey’s very-well-poised $_{6} \psi_{6}$ summation. Bilateral Jackson integral generalizations of the Dixon–Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_{1} \psi_{1}$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulae for Jackson integrals relating to other roots systems obtained from the same perspective.

Let $X$ be a finite set with a continuous action of the absolute Galois group of a global field $K$. We suppose that $X$ is unramified outside a finite set $S$ of places of $K$. For a place $\mathfrak{p} \notin S$, let $N_{X, \mathfrak{p}}$ be the number of fixed points of $X$ by the Frobenius element $\mathrm{Frob}_{\mathfrak{p}} \subset G_{K}$. We define the average value $M(X)$ of $N_{X, \mathfrak{p}}$ where $\mathfrak{p}$ runs through the non-archimedean places in $K$. This generalize the invariant of Chen-Kuan and we apply this for Galois representations. Our results show that there is a certain relationship between $M(X)$ and the size of the image of Galois representations.