In this paper we show some identities come from the \( q \)-identities of Euler, Jacobi, Gauss, and Rogers-Ramanujan. Some of these identities relate the function sum of divisors of a positive integer \( n \) and the number of integer partitions of \( n \). One of the most intriguing results found here is given by the next equation, for \( n > 0 \),
\[
\sum_{l=1}^n \frac{1}{l!} \sum_{w_1+w_2+\cdots+w_l \in C(n)} \frac{\sigma_1(w_1) \sigma_1(w_2) \cdots \sigma_1(w_l)}{w_1 w_2 \cdots w_l} = p_1(n),
\]
where \( \sigma_1(n) \) is the sum of all positive divisors of \( n \), \( p_1(n) \) is the number of integer partitions of \( n \), and \( C(n) \) is the set of integer compositions of \( n \). In the last section, we show seven applications, one of them is a series expansion for
\[
\frac{(q^{a_1};q^{b_1})_\infty (q^{a_2};q^{b_2})_\infty \cdots (q^{a_k};q^{b_k})_\infty}
{(q^{c_1};q^{d_1})_\infty (q^{c_2};q^{d_2})_\infty \cdots (q^{c_r};q^{d_r})_\infty},
\]
where \( a_1, \ldots, a_k, b_1, \ldots, b_k, c_1, \ldots, c_r, d_1, \ldots, d_r \) are positive integers, and \( |q| < 1 \).

Between Bernoulli/Euler polynomials and Pell/Lucas polynomials, convolution sums are evaluated in closed form via the generating function method. Several interesting identities involving Fibonacci and Lucas numbers are shown as consequences including those due to Byrd \( (1975) \) and Frontczak \( (2020) \).

The notion of length spectrum for natural numbers was introduced by Pong in \([5]\). In this article, we answer the question of how often one can recover a random number from its length spectrum. We also include a quick deduction of a result of LeVeque in \([4]\) on the average order of the size of length spectra.

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.