This paper uses exponential sum methods to show that if \( E \subset M_2(\mathbb{Z}/p^r) \)
is a finite set of \( 2 \times 2 \) matrices with sufficiently large density and \( j \) is any unit in the finite ring \( \mathbb{Z}/p^r \), then there exist at least two elements of \( E \) whose difference has determinant \( j \).

In this paper, we introduce a generalized family of numbers and polynomials of one or more variables attached to the formal composition \( f \cdot (g \circ h) \) of generating functions \( f \), \( g \), and \( h \). We give explicit formulae and apply the obtained result to two special families of polynomials; the first concerns the generalization of some polynomials applied to the theory of hyperbolic differential equations recently introduced and studied by \( M. \, Mihoubi \) and \( M. \, Sahari \). The second concerns two-variable Laguerre-based generalized Hermite-Euler polynomials introduced and should be updated to studied recently by \( N. \, U. \, Khan \, \textit{et al.} \).

In this paper, we show that the generalized exponential polynomials and the generalized Fubini polynomials satisfy certain binomial identities and that these identities characterize the mentioned polynomials (up to an affine transformation of the variable) among the class of the normalized Sheffer sequences.

Let \( A \) be a subset of a finite field \( \mathbb{F} \). When \( \mathbb{F} \) has prime order, we show that there is an absolute constant \( c > 0 \) such that, if \( A \) is both sum-free and equal to the set of its multiplicative inverses, then \( |A| < (0.25 – c)|\mathbb{F}| + o(|\mathbb{F}|) \) as \( |\mathbb{F}| \to \infty \). We contrast this with the result that such sets exist with size at least \( 0.25|\mathbb{F}| – o(|\mathbb{F}|) \) when \( \mathbb{F} \) has characteristic 2.

In this paper, we will recover the generating functions of Tribonacci numbers and Chebychev polynomials of first and second kind. By making use of the operator defined in this paper, we give some new generating functions for the binary products of Tribonacci with some remarkable numbers and polynomials. The technique used here is based on the theory of the so-called symmetric functions.

It is shown that if \( V \subseteq \mathbb{F}_p^{n \times \cdots \times n} \) is a subspace of \( d \)-tensors with dimension at least \( tn^{d-1} \), then there is a subspace \( W \subseteq V \) of dimension at least \( t / (dr) – 1 \) whose nonzero elements all have analytic rank \( \Omega_{d, p}(r) \). As an application, we generalize a result of Altman on Szemerédi’s theorem with random differences.

Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these partitions are established. Connections with Riordan arrays are presented.

Every set of natural numbers determines a generating function convergent for \( q \in (-1, 1) \) whose behavior as \( q \to 1^- \) determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set \( D \) of positive integers, call a set \( S \) “\( D \)-avoiding” if no two elements of \( S \) differ by an element of \( D \). We study the problem of determining, for fixed \( D \), all \( D \)- avoiding sets that are maximal in the germ order. In many cases, we can show that there is exactly one such set. We apply this to the study of one-dimensional packing problems.

We prove some combinatorial identities by an analytic method. We use the property that singular integrals of particular functions include binomial coefficients. In this paper, we prove combinatorial identities from the fact that two results of the particular function calculated as two ways using the residue theorem in the complex function theory are the same. These combinatorial identities are the generalization of a combinatorial identity that has been already obtained

Bargraphs are column convex polyominoes, where the lower edge lies on a horizontal axis. We consider the inner site-perimeter, which is the total number of cells inside the bargraph that have at least one edge in common with an outside cell and obtain the generating function that counts this statistic. From this we find the average inner perimeter and the asymptotic expression for this average as the semi-perimeter tends to infinity. We finally consider those bargraphs where the inner site-perimeter is exactly equal to the area of the bargraph.