In this research, we leverage various $ q $-calculus identities to introduce the notion of $ q $-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these $ q $-Hermite-Appell polynomials, encompassing their generating function, series representation, summation equations, recurrence relations, $ q $-differential formula, and operational principles. Our investigation sheds light on the intricate nature of these polynomials, elucidating their behavior and facilitating deeper understanding within the realm of $ q $-calculus.

Various authors have put their sincere efforts into proposing ratio estimators for estimating the population's mean and variance under different situations and sampling methods. But the problem arises when data is unstable, imprecise, ambiguous, incomplete and vague. In such situations, classical methods of estimation do not yield precise results, as these methods are not meant for such problems. Given this difficulty, Neutrosophic statistics are the only alternative as it deals with indeterminacy. So in this study, we have proposed a generalized Neutrosophic robust ratio type estimator which can be used to provide good results in such situations, as well as in the case of the presence of outliers. For the evaluation point of view, we have made use of real data set and simulation study to check the efficacy of our suggested estimators over the mentioned existed estimators.

A mixed family of polynomials, called the Boas-Buck-Sheffer family is introduced and their quasi-monomial properties are established in this article. Also, the generalizations of the Szasz operators including this mixed polynomial family are obtained and their convergence is studied.

This article investigates the properties and monomiality principle within Bell-based Apostol-Bernoulli-type polynomials. Beginning with the establishment of a generating function, the study proceeds to derive explicit expressions for these polynomials, providing insight into their structural characteristics. Summation formulae are then derived, facilitating efficient computation and manipulation. Implicit formulae are also examined, revealing underlying patterns and relationships. Through the lens of the monomiality principle, connections between various polynomial aspects are elucidated, uncovering hidden symmetries and algebraic properties. Moreover, connection formulae are derived, enabling seamless transitions between different polynomial representations. This analysis contributes to a comprehensive understanding of Bell-based Apostol-Bernoulli-type polynomials, offering valuable insights into their mathematical nature and applications.

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