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We give a general identity relating Eisenstein series on general linear groups. We do it by constructing an Eisenstein series, attached to a maximal parabolic subgroup and a pair of representations, one cuspidal and the other a character,…

We prove that the number of $1$-factorisations of a generalised Petersen graph of the type $GP(3k,k)$ is equal to the $k$th Jacobsthal number $J(k)$ if $k$ is odd, and equal to $4J(k)$, when $k$ is even. Moreover, we verify the list…

In this paper we invastigate the notion of generalized (I,J) - Luzin set. This notion generalize the standard notion of Luzin set and Sierpinski set. We find set theoretical conditions which imply the existence of generalized (I,J) - Luzin…

We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we…

In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra $A_{n-1}^{(1)}.$ In…

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial…

Stern's diatomic series, denoted by $(a(n))_{n \geq 0}$, is defined by the recurrence relations $a(2n) = a(n)$ and $a(2n + 1) = a(n) + a(n + 1)$ for $n \geq 1$, and initial values $a(0) = 0$ and $a(1) = 1$. A record-setter for a sequence…

Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and…

The incomplete tribonacci polynomials, denoted by T_n^{(s)}(x), generalize the usual tribonacci polynomials T_n(x) and were introduced in [10], where several algebraic identities were shown. In this paper, we provide a combinatorial…

Suppose that $\alpha,\beta\in\mathbb{R}$. Let $\alpha\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which…

We find generalized Jack polynomials for the group $SU(3)$ and verify that their Selberg averages for several first levels are given by Nekrasov functions. To compute the averages we derive recurrence relations for the $sl_3$ Selberg…

This paper presents a new generalization of the Genocchi numbers and the Genocchi theorem. As consequences, we obtain some important families of integer-valued polynomials those are closely related to the Bernoulli polynomials. Denoting by…

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ satisfying a…

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of…

Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory,…

We extend the definition of Jamison sequences in the context of topological abelian groups. Then we study such sequences when the abelian group is discrete and countably infinite. An arithmetical characterization of such sequences is…

We generalize the notion of trace identity to $J$-trace. Our main result is that all $J$-traces of $M_{n,n}$ are consequence of those of degree $\frac12n(n + 3)$. This also gives an indirect description of the queer trace identities of…

This paper further generalizes a recent result of Shannon and Ollerton who resurrected an old identity due to Ledin. This paper generalizes the Ledin-Shannon-Ollerton result to all the metallic sequences. The results give closed formulas…

Let $M_n(K)$ be the algebra of $n \times n$ matrix over an infinite integral domain $K$. Let $gl_n(K)$ be the Lie algebra of $n \times n$ matrix with the usual Lie product over $K$. Let $G = \{g_1,\ldots,g_n\}$ be a group of order $n$. We…

In this paper, we give recurrence relations and identities for some integer sequences related to Ward numbers such as Ward-Lah numbers, varied Ward numbers and binomial Ward numbers. Most of the sequences are entered in the On-Line…