The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known res...

Gaussian numbers, also called Gaussian polynomials or q-binomial coefficients are the q-analogs of common binomial coefficients. First introduced by Euler these polynomials have played an important role in many different branches of mathematics. Sylvester discovered the unimodality of their coefficients, using invariant theory. Gauss recognized the connection of the coefficients to proper integ...

This is a partial review of the connection theory on differentiable fibre bundles. From different view points, this theory can be found in many works, like [2–6, 9, 13–15, 18, 19, 24, 26, 27, 30, 31, 35–40, 42]. The presentation of the material in Sections 2–5, containing the grounds of the connection theory, follows some of the main ideas of [30, Chapters 1 and 2], but their realization here i...

We consider a Palatini variation on a generalized Einstein-Hilbert action. We find that the Hilbert constraint, that the connection equals the Christoffel symbol, arises only as a special case of this general action, while for particular values of the coefficients of this generalized action, the connection is completely unconstrained. We discuss the relationship between this situation and that ...

We study two sums involving the Stirling numbers and binomial coefficients. We find their closed forms, and discuss the connection between these sums. Dedicated to the memory of our mentors, Professors Leonard Carlitz and Albert Nijenhuis

We prove positivity results about linearization and connection coefficients for Bessel polynomials. The proof is based on a recursion formula and explicit formulas for the coefficients in special cases. The result implies that the distribution of a convex combination of independent Studentt random variables with arbitrary odd degrees of freedom has a density which is a convex combination of cer...