The $q$-Hahn PushTASEP

We introduce the $q$-Hahn PushTASEP—an integrable stochastic interacting particle system that is a three-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (totally asymmetric simple exclusion process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi _3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by Povolotsky [37]. We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. In a $q\to 1$ limit of our process we arrive at a random recursion, which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e., the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner.

A New Class of Functions Suggested by the Generalized Basic Hypergeometric Function

We introduce an extended generalized basic hypergeometric function rΦs+p in which p tends to infinity together with the summation index. We define the difference operators and obtain infinite order difference equation, for which these new special functions are eigen functions. We derive some properties, as the order zero of this function, differential equation involving a particular hyper-Bessel type operators of infinite order, and contiguous function relations. A transformation formula and an l-analogue of the q-Maclaurin's series are also obtained.

A Subclass of Harmonic Univalent Functions Associated with -Analogue of Dziok-Srivastava Operator

We study a class of complex-valued harmonic univalent functions using a generalized operator involving basic hypergeometric function. Precisely, we give a necessary and sufficient coefficient condition for functions in this class. Distortion bounds, extreme points, and neighborhood of such functions are also considered.

Quantum algebra embeddings: deforming functionals and algebraic approach

The study of subalgebras of Lie algebras arising in physical models has been important for many applications. In the present paper we examine the q-deformation of such embeddings; the Lie algebras are then replaced by quantum algebras. Two methods are presented: one based upon deforming functionals, and a direct algebraic approach. A number of examples are given, e.g., [Formula: see text] and [Formula: see text]. For the last example, we give the q-boson construction, and the relevant overlap coefficients are related to a generalized basic hypergeometric function [Formula: see text].