Abstract In this note, we estimate the distance between two q -nomial coefficients ( k n ) q - ? {\left( {_k^n} \right)_q} - {_{k'}^...

In this paper, we consider translation and multiplication operators acting on the rings of symmetric and nonsymmetric polynomials and study their matrix coefficients with respect to the bases of Jack polynomials and interpolation polynomials. The main new insight is that the symmetric and nonsymmetric cases share a key combinatorial feature, that of a locally finite graded poset with a minimum ...

0; n < m for n and m non-negative integers. Binomial coefficients play an important role in many areas of mathematics, including number theory, statistics and probability. Reciprocal binomial coefficients are also prolific in the mathematical literature and many results on reciprocals of binomial coefficient identities may be seen in the papers of Mansour [1], Pla [2], Rockett [3], Sury [5], Su...

There appear to be two standard ways of calculation C-G coefficients in quantum mechanics. One method is to combine the “ladder” operator approach with orthogonality conditions. The second method is to use some type of closed form [2] which allows the coefficients to be calculated directly. However, the latter approach is at times considered “tedious” [2] and does not reveal any new information...

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we show how one can obtain further interesting identities about certain fin...

In the paper, using two inversion theorems for the Stirling numbers and binomial coefficients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two differentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive ...

Abstract. We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities—for example, with all parameters equal to 1—have occurred in the literature. The special case with only ...

As in [1, 2], for rapid numerical calculations of identities pertaining to Lucas or both Fibonacci and Lucas numbers we present each identity as a binomial sum. 1. Preliminaries The two most well-known linear homogeneous recurrence relations of order two with constant coefficients are those that define Fibonacci and Lucas numbers (or Fibonacci and Lucas sequences). They are defined recursively ...

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely proof of this is not known. While proving identity, $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ prime, present formula number subgroups $\mathbb{Z}^s$ finite index $p^r$ with quotient isomorphic to $p$-group type $\underline{\lambda}...