The. theory of reduced incidence algebras of binomial posets furnishes a unified treatment of several types of generating functions that arise in enumerative combinatorics. Using this theory as a tool, we study ‘reduced covering algebras’ of binomial lattices and show that they are isomorphic to various algebras of q-binomial generating functions for certain modular binomial lattices.

We introduce a family of q-analogues of the binomial distribution, which generalises the Stieltjes-Wigert-, Rogers-Szegö-, and Kemp-distribution. Basic properties of this familiy are provided and several convergence results involving the classical binomial, Poisson, discrete normal distribution, and a family of q-analogues of the Poisson distribution are established. These results generalise co...

We prove divisibility properties for sums of powers of binomial coefficients and of q-binomial coefficients. Dedicated to the memory of Paul Erdős

A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.

Assuming the abc-conjecture we show that there are only nitely many powerful binomial coeecients ? n k with 3 k n=2; in fact, that if q 2 divides ? n k that q n 2 ? n k o(1). Unconditionally we show that there are N 1=2+o(1) powerful binomial coeecients in the top N rows of Pascal's Triangle.

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers

Abstract. We start with a (q, t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q, t)-binomial coefficients and their interpretations generalize further in two directions, one relating to col...

Abstract. We start with a (q, t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q, t)-binomial coefficients and their interpretations generalize further in two directions, one relating to col...

Abstract. In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system An. First, we apply Ismail’s [15] argument to an An q-binomial theorem of Milne [25, Th. 5.42] and derive a new An generalization of Ramanujan’s 1ψ1 summation theorem. From this new An 1ψ1 summation and from an An 1ψ1 summation of Gustafson [9] we deduce two lemmas for...