相关论文: The generalized Borwein conjecture. II. Refined q-…
We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an…
We revisit Bressoud's generalized Borwein conjecture. Making use of new positivity-preserving transformations for q-binomial coefficients we establish the truth of infinitely many cases of the Bressoud conjecture. In addition, we prove new…
Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving…
Given an arbitrary ordered pair of coprime integers (a,b) we obtain a pair of identities of the Rogers--Ramanujan type. These identities have the same product side as the (first) Andrews--Gordon identity for modulus 2ab\pm 1, but an…
In this paper, we conjecture an extension to Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state a few infinite hierarchies of non-negative $q$-series identities which are interesting examples of our proposed…
A refinement of the q-trinomial coefficients is introduced, which has a very powerful iterative property. This ``T-invariance'' is applied to derive new Virasoro character identities related to the exceptional simply-laced Lie algebras…
I revisit Bressoud's generalised Borwein conjecture. Making use of certain positivity-preserving transformations for q-binomial coefficients, I establish the truth of infinitely many new cases of the Bressoud conjecture. In addition, I…
We obtain connection coefficients between $q$-binomial and $q$-trinomial coefficients. Using these, one can transform $q$-binomial identities into a $q$-trinomial identities and back again. To demonstrate the usefulness of this procedure we…
We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial…
The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…
The celebrated (First) Borwein Conjecture predicts that for all positive integers~$n$ the sign pattern of the coefficients of the ``Borwein polynomial'' $$(1-q)(1-q^2)(1-q^4)(1-q^5) \cdots(1-q^{3n-2})(1-q^{3n-1})$$ is $+--+--\cdots$. It was…
By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial…
We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the…
Based on a classical result on partitions of an integer into a finite set of positive integers, we establish a general positivity result on coefficients of certain $q$-series which uniformly refines the positivity of truncated pentagonal…
We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain.
Using a property of the q-shifted factorial, an identity for q-binomial coefficients is proved, which is used to derive the formulas for the q-binomial coefficient for negative arguments. The result is in agreement with an earlier paper…
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…
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…
$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…
We appeal to a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultra-distributions we are able to show that the q-Gaussian distribution can be…