Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus

We introduce a bivariate Hahn-factorial q-Laguerre–Tricomi–Appell polynomial class obtained by multiplying a nonsingular Appell factor by a Hahn-factorial deformation of the two-variable q-Laguerre–Tricomi generating kernel. The construction is interpreted formally, coefficientwise, and is not presented as an orthogonality or Laguerre–Hahn functional characterization. Its defining product yields a finite q-binomial convolution, which is the principal mechanism for transferring algebraic and operational properties from the base family to the Appell deformation. We establish direct and inverse connection formulas, a Hessenberg determinant representation, recurrence and higher Hahn-difference formulas, Hahn-shift identities, operational transfer formulas, and quasi-monomiality relations through a basis-defined raising operator. We also derive reductions to the underlying Hahn-factorial q-Laguerre–Tricomi family, one-variable Hahn-Appell and q-Appell families, translated q-Laguerre specializations, admissible Bernoulli–Euler-type subclasses, a singular Genocchi-type convolution, and the joint classical limit q→1−, w→0. A finite coefficient scheme is then used to study representative zero distributions and graphical behavior. The results show that the proposed class is a coherent Appell-type deformation of a Hahn-factorial q-Laguerre–Tricomi kernel and that its structural identities follow from explicitly invertible q-binomial transforms whenever the Appell factor has a nonzero constant term.