In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class ${\mathcal{A}}_p$, where class ${\mathcal{A}}_p$ is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the $(\mathfrak{p},q)$-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter $\mathfrak{p}$ is obviously unnecessary.

Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ${\rho }_e(N,y,x,q)$, and it was noted that ${\rho }_e(\infty ,-1,1,q)$ is the generation of the fifth-order mock theta functions. In the present investigation, several interesting types of generating functions for this q-polynomial using q-difference equations is deduced. Besides that, a generalization of Andrew’s result in form of a multilinear generating function for q-polynomials is also given. Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities. We observe that most of the parameters involved in our results are symmetric to each other. Our results are shown to be connected with several earlier works related to the field of our present investigation.

In terms of a very-well-poised ₆ ϕ ₅ summation formula, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we establish four new q-supercongruences for truncated basic hypergeometric series. One of these results is a new q-analogue of the (F.2) supercongruence of Van Hamme.

In our present investigation, we develop two new Bailey lattices. We describe a number of q-multisums new forms with multiple variables for the basic hypergeometric series which arise as consequences of these two new Bailey lattices. As applications, two new transformations for basic hypergeometric by using the unit Bailey pair are derived. Besides it, we use this Bailey lattice to get some kind of mock theta functions. Our results are shown to be connected with several earlier works related to the field of our present investigation.

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class $S{T}_q(m,c,d,\beta )$. The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties.