# R(p,q)- analogs of discrete distributions: general formalism and application

## TL;DR

This paper introduces a unified framework for $
(p,q)$-deformed discrete distributions, extending classical distributions and linking them to quantum algebra applications, with new formulas for moments, mean, variance, and recursion relations.

## Contribution

It develops a general formalism for $
(p,q)$-deformed distributions, deriving new properties and connecting to quantum algebra, unifying and extending previous results.

## Key findings

- Derived $
(p,q)$-deformed factorial moments and distribution formulas.
- Established recursion relations for $
(p,q)$-distributions.
- Connected $
(p,q)$-deformations to quantum algebra applications.

## Abstract

In this paper, we define and discuss $\mathcal{R}(p,q)$- deformations of basic univariate discrete distributions of the probability theory. We mainly focus on binomial, Euler, P\'olya and inverse P\'olya distributions. We discuss relevant $\mathcal{R}(p,q)-$ deformed factorial moments of a random variable, and establish associated expressions of mean and variance. Futhermore, we derive a recursion relation for the probability distributions. Then, we apply the same approach to build main distributional properties characterizing the generalized $q-$ Quesne quantum algebra, used in physics. Other known results in the literature are also recovered as particular cases.