In this study, we define the unrestricted Pell and Pell-Lucas quaternions. We give generating functions, Binet formulas and some generalizations of well-known identities such as Vajda’s, Catalan’s, Cassini’s d’Ocagne’s identities.

Dust is one of the atmospheric pollutants that have adverse environmental effects and consequences. Dust fall contains particles of 100 microns or even smaller ones, which fall from the atmosphere onto the earth surface. The aim of this study is to determine the concentration of lead in dust fall samples in order to study the pollution level of this element in Zahedan, Sistan and Baluchistan Province, Iran. Therefore, sampling was carried out using 30 marble dust collectors (MDCO) for 3 months in the spring of 2015 to investigate the quantitative variation and spatial analysis of lead content in dust fall. These dust collectors were placed at 30 stations on the building roofs with a height of approximately 1.5 meters across the city. According to the results, the mean lead concentration in the spring was 90.16 mg/kg. In addition, the zoning map of lead content shows that the lowest level of lead was measured at Imam Khomeini station while the highest amount of lead appeared in Mostafa Khomeini station.

The research issue at hand pertains to the intricate mechanisms of state regulation that govern the economy of Kazakhstan, particularly in the context of the international sanctions that have been instituted by the nations comprising the Eurasian Economic Union. In order to thoroughly investigate this complex subject matter, this scholarly paper employs a variety of sophisticated methodologies grounded in bibliometric analyses of the most recent 90 academic papers that focus on the various mechanisms of state regulation pertinent to the economic landscape of Kazakhstan. As a subsequent phase in this research endeavor, the modeling of higher-order moments is undertaken with the express aim of delineating the multifaceted ramifications that stem from a singular and isolated perturbation affecting one of the key variables encapsulated within the higher-order moments model. This detailed analytical approach facilitates an in-depth exploration of both the immediate outcomes and the subsequent values of the endogenous variables that are under scrutiny. The innovative aspect of this article’s findings lies in the comprehensive analysis dedicated to the state regulation of Kazakhstan’s economy, which is significantly influenced by the international sanctions that have been imposed by member countries of the Eurasian Economic Union. The outcomes of this research provide a methodical and scientifically rigorous framework for understanding the overarching system of state regulation, which is of paramount importance for cultivating sustainable development within the socio-economic dynamics that characterize the nation of Kazakhstan.