In this article, we focus our attention on (q,h)-Gauss?s binomial formula from which discover the additive property of (q; h)-exponential functions. We state (q,h)-analogue Gauss?s in terms proper polynomials T(q,h) own essential properties similar to ordinary polynomials. present (q,h)-Taylor series and analyze conditions for its convergence. introduce a new (q,h)-analytic exponential function...

The q-binomial theorem is essentially the expansion of (x − 1)(x − q) · · · (x − q) in terms of the monomials x. In a recent paper [O], A. Okounkov has proved a beautiful multivariate generalization of this in the context of symmetric Macdonald polynomials [M1]. These polynomials have nonsymmetric counterparts [M2] which are of substantial interest, and in this paper we establish nonsymmetric a...

The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads q-Euler integrals and very similar canonical system q-difference equations for functions. q-analogues recurrence formulas in Horns paper Borngässers thesis lead more exact way find these Frobenius solutions. To the right formulas, parameters q-shifted factorials can be changed neg...

The q-binomial coefficients [ n m ] = ∏m i=1(1 − qn−m+i)/(1 − q), for integers 0 ≤ m ≤ n, are known to be polynomials with non-negative integer coefficients. This readily follows from the q-binomial theorem, or the many combinatorial interpretations of [ n m ] . In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of q-facto...

Braided differential operators ∂ are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix R. The quantum eigenfunctions expR(x|v) of the ∂ i (braided-plane waves) are introduced in the free case where the position components xi are totally noncommuti...

In [3] Breuer and Kronholm gave in effect two proofs for an explicit formula for the generating function for partitions where the difference between largest and smallest part is bounded by a given integer t. Their first proof is geometric, involving counting lattice points within a polyhedral region; their second proof constructs an explicit bijection. In this paper we give another proof, a for...

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler’s Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Ch...

We give new q-(1+q)-analogue of the Gaussian coefficient, also know as the q-binomial which, like the original q-binomial [ n k ] q , is symmetric in k and n− k. We show this q-(1 + q)-binomial is more compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our q-(1 + q)-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birk...