We define an overpartition analogue of Gaussian polynomials (also known as q-binomial coefficients) as a generating function for the number of overpartitions fitting inside the M ×N rectangle. We call these new polynomials over Gaussian polynomials or over q-binomial coefficients. We investigate basic properties and applications of over q-binomial coefficients. In particular, via the recurrence...

In this talk we give a survey of results and methods on some combinatorial sums involving binomial coefficients and related to Bernoulli and Euler polynomials. We will also talk about certain sums of minima and maxima related to Dedekind sums. Some interesting identities associated with the various sums will also be introduced. 1. A curious identity and the sum ∑ k≡r (mod m) ( n k ) In 1988 Zhi...

An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are h-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic a...

The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationsh...

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence { ∑n k=0 ( n k )2 q}n≥0. By using Pieri’s rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also pr...

Setting b = 0 in (1) produces the binomial identity (henceforth BI), which explains Abel’s interest in AI as a generalization of the BI. Unfortunately, in later expositions [3, 4], AI is usually presented with the fixed choice b = -1, thus severing the link between the two identities. In (1) the symbols ,  appearing in the original text were replaced with a, b, respectively, and the current s...

A tile composed of two pieces, which we refer to as a fence tile, is introduced to give the Tribonacci numbers a new combinatorial interpretation. The interpretation is used to construct a Pascal-like triangle and various identities concerning the triangle are proven. An intuitive proof of a general identity for the Tribonacci numbers in terms of sums of products of the binomial coefficients is...

The main results of this paper are computations of a few determinants related to binomial coefficients. The most interesting example, discovered after browsing through [5], is obtained by considering binomial coefficients modulo 4. We include related results contained in [2] and [3]. The next section recalls mostly well-known facts concerning binomial and q−binomial coefficients and states the ...

In this paper, we have proposed a novel VLSI-oriented approach to computing the rotation matrix entries from the quaternion coefficients. The advantage of this approach is the complete elimination of multiplications and replacing them by less costly squarings. Our approach uses Logan's identity, which proposes to replace the calculation of the product of two numbers on summing the squares via t...