Related papers: On an identity by Chaundy and Bullard. I
Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of…
The aim of this research paper is to obtain explicit expressions of (i) $ {}_1F_1 \left[\begin{array}{c} \alpha \\ 2\alpha + i \end{array} ; x \right]. {}_1F_1\left[ \begin{array}{c} \beta \\ 2\beta + j \end{array} ; x \right]$ (ii)…
In this note we present a combinatorial proof of an identity involving poly-Bernoulli numbers and Genocchi numbers. We introduce the combinatorial objects, $m-$barred Callan sequences and show that the identity holds in a more general…
A well-known and fundamental property of the Macdonald polynomials $P_\lambda(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. Recently, Concha and Lapointe showed that this property extends in an…
In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem, which opened up a new study on truncated theta series. In particular, some truncated versions of a identity of Gauss have been proved. In this…
The multiplicate form of Gould--Hsu's inverse series relations enables to investigate the dual relations of the Chu-Vandermonde-Gau{\ss}'s, the Pfaff-Saalsch\"utz's summation theorems and the binomial convolution formula due to Hagen and…
In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, $$ \sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1}, $$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$. Recently,…
In this paper, by the technique of inverse relations and comparing coefficients, we establish some generalized forms of Andrews' q-series identity and two new Bailey pairs and q-identities closely related to Andrews-Warnaar's sum identity…
The first results presented in our article are the clear definitions of both intrinsic and extrinsic discrete curvatures in terms of holonomy and plane-angle representation, a clear relation with their deficit angles, and their clear…
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…
The multi-vector generalization of a rigid, partially-broken $\mathcal{N}=2$ supersymmetric theory is presented as a rigid limit of a suitable gauged $\mathcal{N}=2$ supergravity with electric, magnetic charges and antisymmetric tensor…
In this paper, we prove two related central binomial series identities: $B(4)=\sum_{n \geq 0} \frac{\binom{2n}n}{2^{4n}(2n+1)^3}=\frac{7 \pi^3}{216}$ and $C(4)=\sum_{n \in \mathbb{N}} \frac{1}{n^4 \binom{2n}n}=\frac{17 \pi^4}{3240}.$ Both…
Let $s_{k}(n)$ denote the sum of digits of an integer $n$ in base $k$. Motivated by certain identities of Nieto, and Bateman and Bradley involving sums of the form $\sum_{i=0}^{2^{n}-1}(-1)^{s_{2}(i)}(x+i)^{m}$ for $m=n$ and $m=n+1$, we…
In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and…
The title identity appeared as Problem 75-4, proposed by P. Barrucand, in Siam Review in 1975. The published solution equated constant terms in a suitable polynomial identity. Here we give a combinatorial interpretation in terms of card…
Based on Jensen formulae and the second kind of Chebyshev polynomials, another proof is presented for an extension of a curious binomial identity due to Z. W. Sun and K. J. Wu.
We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$…
Ramanujan wrote the following identity \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 +…
Inspired by two entries published in Ramanujan's lost notebook on Page 355, B. C. Berndt et al.\cite{MR3351542} presented Riesz sum identities for Ramanujan entries by introducing the twisted divisor sums. Later, S. Kim \cite{MR3541702}…
We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also…