Binomial coefficients of the form \( \binom{\alpha}{\beta} \) for complex numbers \( \alpha \) and \( \beta \) can be defined in terms of the gamma function, or equivalently the generalized factorial function. Less well-known is the fact that if \( n \) is a natural number, the binomial coefficient \( \binom{n}{x} \) can be defined in terms of elementary functions. This enables us to investigate the function \( \binom{n}{x} \) of the real variable \( x \). The results are completely in line with what one would expect after glancing at the graph of \( \binom{3}{x} \), for example, but the techniques involved in the investigation are not the standard methods of calculus. The analysis is complicated by the existence of removable singularities at all of the integer points in the interval \( [0, n] \), and requires multiplying, rearranging, and differentiating infinite series.

In this paper, we analyze the stochastic properties of some large size (area) polyominoes’ perimeter such that the directed column-convex polyomino, the columnconvex polyomino, the directed diagonally-convex polyomino, the staircase (or parallelogram) polyomino, the escalier polyomino, the wall (orbargraph) polyomino. All polyominoes considered here are made of contiguous, not-empty columns, without holes, such that each column must be adjacent to some cell of the previous column. We compute the asymptotic (for large size n) Gaussian distribution of the perimeter, including the corresponding Markov property of the chain of columns, and the convergence to classical Brownian motions of the perimeter seen as a trajectory according to the successive columns. All polyominoes of size n are considered as equiprobable.

Convolution conditions are discussed for the \(q\)-analogue classes of Janowski starlike, convex and spirallike functions.

we discuss a framework for constructing large subsets of \(\mathbb{R}^n\) and \(K^n\) for non-archimedean local fields \(K\). This framework is applied to obtain new estimates for the Hausdorff dimension of angle-avoiding sets and to provide a counterexample to a limiting version of the Capset problem.

Let \( R \) be a commutative ring with unity and \( M \) be an \( R \)-module. The total graph of \( M \) with respect to the singular submodule \( Z(M) \) of \( M \) is an undirected graph \( T(\Gamma(M)) \) with vertex set as \( M \) and any two distinct vertices \( x \) and \( y \) are adjacent if and only if \( x + y \in Z(M) \). In this paper, the author attempts to study the domination in the graph \( T(\Gamma(M)) \) and investigate the domination number and the bondage number of \( T(\Gamma(M)) \) and its induced subgraphs. Some domination parameters of \( T(\Gamma(M)) \) are also studied. It has been shown that \( T(\Gamma(M)) \) is excellent, domatically full, and well covered under certain conditions.

In this paper, we study a class of sequences of polynomials linked to the sequence of Bell polynomials. Some sequences of this class have applications on the theory of hyperbolic differential equations and other sequences generalize Laguerre polynomials and associated Lah polynomials. We discuss, for these polynomials, their explicit expressions, relations to the successive derivatives of a given function, real zeros and recurrence relations. Some known results are significantly simplified.

We show that if \( G \) is a discrete Abelian group and \( A \subseteq G \) has \( \|1_A\|_{B(G)} \leq M \), then \( A \) is \( O(\exp(\pi M)) \)-stable in the sense of Terry and Wolf.