The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota’s results on the author’s research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differentia equations.

Let S be a semigroup. Our main result is that we describe the complex-valued solutions of the following functional equations
g(xσ(y)) = g(x)g(y) + f(x)f(y), x, y ∈ S,
f(xσ(y)) = f(x)g(y) + f(y)g(x), x, y ∈ S,
and
f(xσ(y)) = f(x)g(y) - f(y)g(x), x, y ∈ S,
where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants.
We also give some applications.

In this paper we are focusing on finding the transcendental entire solution of Fermat-type trinomial and binomial equations, by restricting the hyper-order to be less than one. As the hyper-order is a crucial parameter that characterizes the growth of entire functions, it will be interesting to investigate this unexplored domain, as far as practible, with certain restriction on hyper order. Our results are the improvements of previous results reported in recent papers [12], [13]. We have provided a series of examples to demonstrate and validate the effectiveness of our proposed solutions.

We present Hermite-Hadamard-Fejér type inequalities for strongly M_φM_ψ-convex functions. Some refinements of them and bounds for the integral mean of the product of two functions are also obtained.

In this paper we consider a generalized polynomial f : ℝ → ℝ of degree two that satisfies the additional equation f(x)f(y) = 0 for the pairs (x,y) ∈ D, where D ⊆ ℝ² is given by some algebraic condition. In the particular cases when there exists a positive rational m fulfilling
D = { (x,y) ∈ ℝ² | x² - my² = 1 },
we prove that f(x) = 0 for all x ∈ ℝ.

In this paper, we define and study a new one-parameter generalization of the Pell hybrid numbers. Based on the definition of r-Pell numbers, we define the r-Pell hybrid numbers. We give their properties: character, Binet formula, summation formula, and generating function. Moreover, we present Catalan, Cassini, d’Ocagne, and Vajda type identities for the r-Pell hybrid numbers.

A new bidimensional version of cobalancing numbers and Lucasbalancing numbers are introduced. Some properties and identities satisfied by these new bidimensional sequences are studied.

In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks.

Let u_n = u_n(k) denote the generalized Leonardo number defined recursively by u_n = u_n-1 + u_n-2 + k for n≥2, where u₀ = u₁ = 1. Terms of the sequence u_n(1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1≤k≤7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k=1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.

We prove a theorem which unifies some formulas, for example the counting function of some sets of numbers including all positive integers, h-free numbers, h-full numbers, etc. We also establish a conjecture and give some examples where the conjecture holds.

In this paper we study a family of doubled and quadrupled Fibonacci type sequences obtained by distance generalization of Fibonacci sequence. In particular we obtain doubled Fibonacci sequence, doubled and quadrupled Padovan sequence and quadrupled Narayana’s sequence. We give a binomial direct formula for these sequences using graph methods, and also we derive a number of identities. Moreover, we study matrix generators of these sequences and determine connections with the Pascal’s triangle.

In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra L is used to study the unifying properties of some subclasses of the lattice of filters of L. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored.

Report of Meeting. The Twenty-third Katowice-Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 - February 3, 2024.