The XIII Annual Lecture dedicated to the memory of Professor Andrzej Lasota

The aim of this paper is to characterize the solutions Φ:G→M₂(ℂ) of the following matrix functional equations
\frac{Φ(xy)+Φ(σ(y)x)}{2} = Φ(x)Φ(y), x,y∈G,
and
\frac{Φ(xy)-Φ(σ(y)x)}{2} = Φ(x)Φ(y), x,y∈G,
where G is a group that need not be abelian, and σ:G→G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

The aim of this note is to study the distribution function of certain sequences of positive integers, including, for example, Bell numbers, factorials and primorials. In fact, we establish some general asymptotic formulas in this regard. We also prove some limits that connect these sequences with the number e. Furthermore, we present a generalization of the primorial.

In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F(x) = σx+... is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation.

If (μ_n)_n=1^∞ are positive measures on a measurable space (X,Σ) and (v_n)_n=1^∞ are elements of a Banach space (X,Σ) such that Σ_n=1^∞‖v_n‖μ_n(X)<∞, then ω(S) = Σ_n=1^∞v_nμ_n(S) defines a vector measure of bounded variation on (X,Σ). We show E has the Radon-Nikodym property if and only if every E-valued measure of bounded variation on (X,Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.
We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on E-valued measures for any Banach space E that has the Radon-Nikodym property.

A generalization of the Hermite–Hadamard (HH) inequality for a positive convex stochastic process, by means of a newly proposed fractional integral operator, is hereby established. Results involving the Riemann–Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl and Liouville fractional integrals are deduced as particular cases of our main result. In addition, we also apply some known HH results to obtain some estimates for the expectations of integrals of convex and p-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral, Revista Integración, temas de matemáticas 36 (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction.

A new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P_n^l is expressed as a linear combination of P_mn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R=2^r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

In this paper, we introduce the generalized Tetranacci hybrid numbers and, as special cases, Tetranacci and Tetranacci-Lucas hybrid numbers. Moreover, we present Binet’s formulas, generating functions, and the summation formulas for those hybrid numbers.