The Q-Charlier polynomials are a family of orthogonal polynomials that arise in the context of probability and combinatorial analysis. They are a specific case of the Charlier polynomials, which are defined concerning Poisson distribution. The Q-Charlier polynomials extend this concept to the setting of the \( q \)-calculus, which incorporates a parameter \( q \) that allows for generalization and flexibility in combinatorial structures.

The Q-Hahn polynomials are a family of orthogonal polynomials that arise in the context of basic hypergeometric functions and q-series. They are a specific case of the more general class of q-polynomials, which are related to the theory of partition and combinatorics, as well as to special functions in mathematical physics.

The Q-Krawtchouk polynomials are a set of orthogonal polynomials that generalize the Krawtchouk polynomials, which themselves are a class of discrete orthogonal polynomials. The Krawtchouk polynomials arise in combinatorial settings and are connected to binomial distributions, while the Q-Krawtchouk polynomials introduce a parameter \( q \) that allows for further generalization. ### Definition and Properties 1.

Q-Laguerre polynomials are a generalization of the classical Laguerre polynomials that arise in quantum mechanics and mathematical physics. They are part of the family of orthogonal polynomials, and they can be associated with various applications, including the study of quantum harmonic oscillators, wave functions of certain quantum systems, and in numerical analysis.

Q-Racah polynomials are a class of orthogonal polynomials that arise in the context of the theory of special functions and are associated with the asymptotic theory of orthogonal polynomials. They are a generalization of the Racah polynomials and belong to the family of basic hypergeometric orthogonal polynomials.

Quantum \( q \)-Krawtchouk polynomials are a family of orthogonal polynomials that can be seen as a \( q \)-analogue of the classical Krawtchouk polynomials. They arise in various areas of mathematics, particularly in the theory of quantum groups, representation theory, and combinatorial analysis. ### Definitions and Properties 1.

Perpendicular distance refers to the shortest distance from a point to a line, plane, or a geometric shape. This distance is measured along a line that is perpendicular (at a 90-degree angle) to the surface or line in question. ### Key Points: - **From a Point to a Line**: The perpendicular distance from a point to a line is the length of the segment that connects the point to the line at a right angle.

OS/2 is an operating system developed by IBM and Microsoft that was introduced in the late 1980s. It features a command-line interface similar to DOS and includes a set of commands that can be used to perform various tasks, manage files, and control system functions. Here are some common OS/2 commands: 1. **DIR** - Displays a list of files and directories in the specified directory. - Example: `DIR C:\` 2.

Turán's inequalities refer to a set of inequalities related to the sums of powers of sequences of real numbers. These inequalities are particularly significant in the context of polynomial approximations and the theory of symmetric polynomials.

The term "perpendicular" refers to the relationship between two lines, segments, or planes that meet or intersect at a right angle (90 degrees). In two-dimensional geometry, if line segment \( AB \) is perpendicular to line segment \( CD \), it means they intersect at an angle of 90 degrees. In three-dimensional space, the concept extends similarly; for example, a line can be said to be perpendicular to a plane if it intersects the plane at a right angle.

In group theory, which is a branch of abstract algebra, a **P-group** is a type of group that plays an important role in the study of finite groups. Specifically, a P-group is defined as a group in which the order (the number of elements) of every element is a power of a prime number \( p \).

Bin packing is a type of combinatorial optimization problem that involves packing a set of items of varying sizes into a finite number of bins or containers in such a way that minimizes the number of bins used. The objective is to efficiently utilize space (or capacity) while ensuring that the items fit within the constraints of the bins. ### Key Concepts 1. **Items**: Each item has a specific size or weight. 2. **Bins**: Each bin has a maximum capacity that cannot be exceeded.

In geometry, the term "normal" can refer to several concepts, but it is most commonly used in relation to the idea of a line or vector that is perpendicular to a surface or another line. Here are a few contexts in which "normal" is used: 1. **Normal Vector:** In three-dimensional space, a normal vector to a surface at a given point is a vector that is perpendicular to the tangent plane of the surface at that point.

OS/2 is an operating system that was originally developed by IBM and Microsoft in the late 1980s. It was designed to be a robust, multitasking operating system for personal computers, especially for business and enterprise use. Although Microsoft eventually exited the OS/2 project to focus on Windows, IBM continued to develop OS/2 into the 1990s.

OS/2, short for Operating System/2, is an operating system developed by IBM in collaboration with Microsoft in the late 1980s. It was initially designed as a successor to DOS and intended to be a more advanced platform for personal computing. The OS/2 operating system featured a graphical user interface and was known for its multitasking capabilities, stability, and support for running multiple applications simultaneously. OS/2 went through several versions, with notable releases including OS/2 1.

OS/2, which stands for Operating System/2, is a computer operating system developed by IBM in the late 1980s. Originally created as a successor to DOS, OS/2 was intended to provide a stable and capable environment for running applications in a multitasking and multiuser setting. It was co-developed with Microsoft initially, but after version 1.3, IBM took over development completely.

OS/2 drivers are software components that allow the OS/2 operating system to communicate with hardware devices and facilitate their functioning. OS/2, developed by IBM, is a multi-tasking operating system that was originally designed for personal computers, and it supports a variety of hardware components, including printers, network cards, storage devices, and graphics adapters.

Windows Libraries for OS/2 (WL/2) was a software package developed by IBM that allowed certain Windows applications to run on the OS/2 operating system. Released in the early 1990s, it provided a compatibility layer that facilitated the execution of 16-bit Windows applications, effectively enabling users to take advantage of the growing library of Windows software while using OS/2 as their primary operating system.

Ulam's packing conjecture is a hypothesis in the field of geometry and combinatorial mathematics, particularly concerning the arrangement of spheres in space. Formulated by mathematician Stanislaw Ulam, the conjecture posits that the densest packing of spheres (in three-dimensional space) occurs when the spheres are arranged in a face-centered cubic (FCC) lattice structure or equivalently in a hexagonal close packing (HCP) arrangement.

The 20th century saw significant contributions from Pakistani mathematicians, particularly in the context of the country's formation and its growth in higher education and research. Here are some notable mathematicians from Pakistan during that time: 1. **Abdul Salam**: Although primarily known for his work in theoretical physics, Abdul Salam also made significant contributions to mathematical physics. He was awarded the Nobel Prize in Physics in 1979. 2. **Muhammad G.