We present a new method to compute formulas for the action on monomials of a generalization of binomial approximation operators of Popoviciu type, or equivalently moments of associated discrete probability distributions with finite support. These quantities are necessary to check the assumptions of the Korovkin Theorem for approximation operators, or equivalently the Feller Theorem for converge...

Abstract. We investigate a (q, t)-generalization of the usual binomial and q-binomial coefficients. 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 polynomials and their interpretations generalize further in two directions, one relatin...

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 column-strict...

In the present paper combinatorial identities involving q-dual sequences or polynomials with coefficients q-dual sequences are derived. Further, combinatorial identities for q-binomial coefficients(Gaussian coefficients), q-Stirling numbers and q-Bernoulli numbers and polynomials are deduced.

The author introduces overpartitions and particle seas as a generalization of partitions. Both new tools are used in bijective proofs of basic hypergeometric identities like the q-binomial theorem, Jacobi’s triple product, q-Gauß equality or even Ramanujan’s 1Ψ1 summation. 1. Partitions In 1969, G. E. Andrews was already looking for bijective proofs for some basic hypergeometric identities. The...

The main idea is to consider generalized binomial coefficients that are formed from an arbitrary sequence C, as shown in (3) below. We will isolate a property of the sequence C that guarantees the existence of a theorem like Kummer’s, relating divisibility by prime powers to carries in addition. A special case of the theorem we shall prove describes the prime power divisibility of Gauss’s gener...