Undergraduate Students’ Combinatorial Proof of Binomial Identities

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

A Note on the Difference of Powers and Falling Powers

Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. They can be established by many techniques, from generating functions to special series. Here, using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. This way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. The findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics.

TWO SUPERCONGRUENCES RELATED TO MULTIPLE HARMONIC SUMS

Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form $$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$ which generalise previous results. We also establish q-analogues of two binomial identities.

Evaluation of sums involving products of Gaussian q-binomial coefficients with applications

Sums of products of two Gaussian q-binomial coefficients, are investigated, one of which includes two additional parameters, with a parametric rational weight function. By means of partial fraction decomposition, first the main theorems are proved and then some corollaries of them are derived. Then these q-binomial identities will be transformed into Fibonomial sums as consequences.

Combinatorial identities involving the central coefficients of a Sheffer matrix

Given m ? N, m ? 1, and a Sheffer matrix S = [sn,k]n,k?0, we obtain the exponential generating series for the coefficients (a+(m+1)n a+mn)-1 sa+(m+1)n,a+mn. Then, by using this series, we obtain two general combinatorial identities, and their specialization to r-Stirling, r-Lah and r-idempotent numbers. In particular, using this approach, we recover two well known binomial identities, namely Gould's identity and Hagen-Rothe's identity. Moreover, we generalize these results obtaining an exchange identity for a cross sequence (or for two Sheffer sequences) and an Abel-like identity for a cross sequence (or for an s-Appell sequence). We also obtain some new Sheffer matrices.

Identities for Harmonic Numbers and Binomial Relations Via Legendre Polynomials

We employ the orthonormality of the Legendre polynomials to deduce binomial identities. The harmonic numbers Hn are connected with the derivatives of binomial coefficients, this fact allows to deduce identities involving the Hn.Kathmandu University Journal of Science, Engineering and TechnologyVol. 13, No. 2, 2017, page: 92-97