The quasi frame is more efficient than the Frenet frame in investigating surfaces, and it is regarded a generalization frame of both the Frenet and Bishop frames. The geometry of quasi-Hasimoto surfaces in Minkowski 3-space ${\mathbb{E}}_1^3$ is investigated in this paper. For the three situations of non-lightlike curves, the geometric features of the quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ are examined and the Gaussian and mean curvatures for each case are determined. The quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ must satisfy a necessary and sufficient condition to be developable surfaces. As a result, the parameter curves of quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ is described. Thus, the $s$-parameter and $t$-parameter curves of quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ are said to be geodesics, asymptotic, and curvature lines under necessary and sufficient circumstances are proved. Finally, quasi curves and associated quasi-Hasimoto surface correspondences are discussed.

In this paper, we consider a new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type, denoted by ${\mathcal{W}}_{n,\lambda }^{(\alpha )}(\delta ,\zeta ;\rho ;\mu )$. We obtain several summation formulae, a recurrence relation, two difference operator formulas, two derivative operator formulas, an implicit summation formula, and a symmetric property for these polynomials. Also, we provide a representation of the degenerate differential operator on the degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type. Moreover, we define the degenerate unified Hermite-based Apostol-Stirling polynomials of the second kind and derive some properties of these newly established polynomials. In addition, we prove multifarious correlations, including the new polynomials. Furthermore, we list the first few degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type for some special cases and present data visualizations of zeros forming 2D and 3D structures. Finally, we provide a table covering approximate solutions for the zeros of ${\mathcal{W}}_{n,3}^{(\alpha )}(\delta ,4;3;2)$.

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.

This paper presents a geometric study of three types of ruled surfaces generated from the tangent, normal, and binormal unit vectors of unit speed space curves. Using the T-pedal curve construction as a foundation, we analyze these surfaces through their fundamental geometric forms, including curvature properties, the striction curve geometry, and the distribution parameter. The theoretical framework is used to analyze problems in computational geometry and shape modeling, with results relevant to both mathematical research and engineering applications. The work establishes fundamental geometric insights while providing tools for applied shape modeling and analysis.