Polynomials

Polynomials are mathematical expressions that consist of variables (often represented by letters) and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.

Generating functions are a powerful mathematical tool used in combinatorics, probability, and other areas of mathematics to encode sequences of numbers into a formal power series. Essentially, a generating function provides a way to express an infinite sequence as a single entity, allowing for easier manipulation and analysis.

In algebraic topology, Betti numbers are a sequence of integers that provide important information about the topology of a topological space. They are used to classify spaces based on their connectivity properties and to understand their shape and structure. Specifically, the \(n\)-th Betti number, denoted \(b_n\), represents the rank of the \(n\)-th homology group \(H_n(X)\) of a topological space \(X\).

Cyclic sieving is a concept from combinatorics, particularly in the area of enumerative combinatorics, which relates to counting combinatorial objects using the cycle structure of permutations. The main idea behind cyclic sieving is to understand how a family of combinatorial objects can be partitioned or "sieved" based on the action of a finite group, particularly the cyclic group.

The factorial moment generating function (FMGF) is a generating function that is particularly useful in probability and statistics for dealing with discrete random variables, especially those that take non-negative integer values. The FMGF is closely related to the moments of a random variable but is structured in a way that makes it suitable for analyzing distributions where counts or frequencies are relevant, like the Poisson distribution or the negative binomial distribution.

A generating function is a formal power series whose coefficients encode information about a sequence of numbers or combinatorial objects. It is a powerful tool in combinatorics and other fields of mathematics because it provides a way to manipulate sequences algebraically.

Generating function transformation refers to a mathematical technique used in combinatorics and related fields that involves the use of generating functions to study sequences, count combinatorial objects, or solve recurrence relations. A generating function is a formal power series in one or more variables, where the coefficients of the series correspond to terms in a sequence. ### Types of Generating Functions 1.

Matsushima's formula is used in the field of celestial mechanics and astrophysics, particularly in the context of estimating the gravitational influence of celestial bodies on the orbits of other objects. It provides a way to calculate the potential influence of a source mass on the motion of surrounding objects. The formula is often expressed in terms of the gravitational potential or force acting on an object due to a celestial body, taking into account both the mass of the body and its distance from the object in question.

The moment-generating function (MGF) is a mathematical tool used in probability theory and statistics to characterize the distribution of a random variable. It is defined as the expected value of the exponential function of the random variable.

A probability-generating function (PGF) is a specific type of power series that is used to encode the probabilities of a discrete random variable. It is particularly useful in the study of probability distributions and in solving problems involving sums of independent random variables. ### Definition For a discrete random variable \( X \) that takes non-negative integer values (i.e.

The Tau function is an important concept in the study of integrable systems, particularly in the context of algebraic geometry, mathematical physics, and soliton theory. It serves as a generating function that encodes information about the solutions to certain integrable equations, such as the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, or the Toda lattice.

Weisner's method is a systematic approach used in number theory to derive new results or solve problems about Diophantine equations, which are polynomial equations that seek integer solutions. Named after the mathematician Boris Weisner, the method emphasizes using algebraic manipulation and properties of integers to explore and generate solutions. One common application of Weisner's method is in the context of Pell's equation, where particular techniques can help identify solutions or transformations that simplify the equation.

Homogeneous polynomials are a special class of polynomials that have the property that all their terms have the same total degree. In mathematical terms, a polynomial \( P(x_1, x_2, \ldots, x_n) \) is considered homogeneous of degree \( d \) if every term in the polynomial is of degree \( d \).

Quadratic forms are expressions involving a polynomial of degree two in several variables.

The complete homogeneous symmetric polynomial is a fundamental concept in algebra, particularly in the theory of symmetric functions.

Diagonal form refers to a way of representing matrices or linear transformations that simplifies the analysis and computation of systems of equations. Specifically, a matrix is said to be in diagonal form when all of its non-zero elements are located along its main diagonal, and all other elements are zero.

Elementary symmetric polynomials are a fundamental class of symmetric polynomials in algebra. Given a set of \( n \) variables, \( x_1, x_2, ..., x_n \), the elementary symmetric polynomials are defined as follows: 1. The first elementary symmetric polynomial \( e_1(x_1, x_2, ...

Polynomial SOS (Sum of Squares) refers to a specific class of polynomial expressions that can be represented as a sum of squares of other polynomials.

Power sum symmetric polynomials are a specific type of symmetric polynomial that represent sums of powers of the variables.

SOS-convexity, or Sum of Squares convexity, is a concept in optimization and mathematical programming that relates to certain types of convex functions. A function is said to be SOS-convex if it allows for a polynomial representation that can be described using sums of squares.

The Schur polynomial is a specific type of symmetric polynomial that plays a significant role in algebraic combinatorics, representation theory, and geometry. It is associated with a given partition of integers and is used in the study of symmetric functions.

Orthogonal polynomials are a class of polynomials that satisfy specific orthogonality conditions with respect to a given weight function over a certain interval.

Affine \( q \)-Krawtchouk polynomials are a family of orthogonal polynomials that arise in the context of quantum calculus or non-classical orthogonal polynomial theory, particularly in relation to \( q \)-analogs of established mathematical concepts. These polynomials generalize the classical Krawtchouk polynomials, which are associated with the binomial distribution and combinatorial problems.

An affine root system is an extension of the concept of root systems, which are used in the theory of Lie algebras and algebraic groups. The affine root system is associated with affine Lie algebras, which are a class of infinite-dimensional Lie algebras that arise in the study of symmetries and integrable systems.

Al-Salam–Carlitz polynomials are a family of orthogonal polynomials that generalize the classical Carlitz polynomials. They appear in the context of q-series and combinatorial identities and are related to various areas in mathematics, including number theory and formal power series. These polynomials are typically defined in terms of parameters \( a \) and \( b \) and a variable \( x \).

The Al-Salam–Chihara polynomials are a family of orthogonal polynomials that arise in the theory of special functions, specifically in the context of q-series and quantum calculus. They are named after the mathematicians Abd al-Rahman Al-Salam and Jun-iti Chihara, who contributed to their study.

The Askey scheme is a classification of orthogonal polynomial sequences that arise in the context of special functions and approximation theory. Named after Richard Askey, this scheme organizes orthogonal polynomials into a hierarchy based on their properties and relationships.

The Askey–Gasper inequality is a result in the field of mathematical analysis, particularly in the study of special functions and orthogonal polynomials. It provides bounds for certain types of sums and integrals involving orthogonal polynomials, especially within the context of Jacobi polynomials.

Associated Legendre polynomials are a generalization of Legendre polynomials, which arise in the context of solving problems in physics, particularly in potential theory, quantum mechanics, and in the theory of spherical harmonics. The associated Legendre polynomials, denoted as \( P_\ell^m(x) \), are defined for non-negative integers \( \ell \) and \( m \), where \( m \) can take on values from \( 0 \) to \( \ell \).

Bateman polynomials, named after the mathematician Harry Bateman, are a family of orthogonal polynomials that arise in various contexts in mathematics, particularly in the theory of special functions and approximation theory. They are often denoted by \( B_n(x) \) and defined using a specific recurrence relation or via their generating functions.

Bessel polynomials are a series of orthogonal polynomials that are related to Bessel functions, which are solutions to Bessel's differential equation. The Bessel polynomials, denoted usually by \( P_n(x) \), are defined using the formula: \[ P_n(x) = \sum_{k=0}^{n} \binom{n}{k} \frac{(-1)^k}{k!} (x/2)^k.

Big \( q \)-Laguerre polynomials are a specific family of orthogonal polynomials that arise in the context of \( q \)-analysis, a generalization of classical analysis that incorporates the parameter \( q \). These polynomials are particularly useful in various areas of mathematics and mathematical physics, including quantum calculus, combinatorics, and orthogonal polynomial theory.

Biorthogonal polynomials are a generalization of orthogonal polynomials where two different systems of polynomials are orthogonal with respect to two different measures.

The Christoffel–Darboux formula is a significant result in the theory of orthogonal polynomials. It provides a way to express sums of products of orthogonal polynomials in a concise form. Typically, the formula relates the orthogonal polynomials defined on a specific interval with respect to a weight function.

Classical orthogonal polynomials are a set of orthogonal polynomials that arise in various areas of mathematics, especially in the context of approximation theory, numerical analysis, and mathematical physics. These polynomials are defined on specific intervals and with respect to certain weight functions, leading to their orthogonality properties.

Continuous Hahn polynomials are a family of orthogonal polynomials that arise in the context of approximation theory and quantum physics. They are part of the broader family of hypergeometric orthogonal polynomials and are linked to various mathematical fields, including special functions, approximation theory, and the theory of orthogonal polynomials.

Continuous big \( q \)-Hermite polynomials are a family of orthogonal polynomials that arise in the study of special functions, particularly in the context of quantum calculus or \( q \)-analysis. They are part of the wider family of \( q \)-orthogonal polynomials, which generalize classical orthogonal polynomials by introducing a parameter \( q \).

The continuous dual Hahn polynomials are a family of orthogonal polynomials that arise in the context of special functions and quantum calculus. They are part of the broader family of dual Hahn polynomials and have applications in various areas, including mathematical physics, combinatorics, and approximation theory. The continuous dual Hahn polynomials can be defined in terms of a three-parameter family of polynomials, which can be specified using recurrence relations or generating functions.

Continuous dual \( q \)-Hahn polynomials are a family of orthogonal polynomials that arise in the context of basic hypergeometric series and quantum group theory. They are a part of the \( q \)-Askey scheme, which organizes various families of orthogonal polynomials based on their properties and connections to special functions.

Continuous \( q \)-Hahn polynomials are a class of orthogonal polynomials that arise in the study of special functions, particularly in the context of \( q \)-series and quantum groups. They are a part of a broader family of \( q \)-analogues of classical orthogonal polynomials, which includes the \( q \)-Hahn, \( q \)-Jacobi, and others.

Continuous q-Hermite polynomials are a set of orthogonal polynomials that arise in the context of q-calculus and are related to various areas in mathematics and physics, especially in the theory of special functions and quantum groups. They are a q-analogue of the classical Hermite polynomials. ### Definition and Properties 1.

Continuous q-Jacobi polynomials are a family of orthogonal polynomials that generalize the classical Jacobi polynomials in the context of q-analogs, which are important in various areas of mathematics, including combinatorics, number theory, and quantum calculus.

Continuous \( q \)-Laguerre polynomials are a family of orthogonal polynomials that generalize the classical Laguerre polynomials by incorporating the concept of \( q \)-calculus, which deals with discrete analogs of calculus concepts. These polynomials arise in various areas of mathematics and physics, including approximation theory, special functions, and quantum mechanics.

Discrete Chebyshev polynomials are a sequence of orthogonal polynomials defined on a discrete set of points, typically related to the Chebyshev polynomials of the first kind. These discrete polynomials arise in various applications, including numerical analysis, approximation theory, and computing discrete Fourier transforms. The discrete Chebyshev polynomials are defined based on the characteristic roots of the Chebyshev polynomials, which correspond to specific points on an interval.

Discrete orthogonal polynomials are a class of polynomials that are orthogonal with respect to a discrete measure or inner product. This means that they are specifically defined for sequences of points in a discrete set (often integers or specific values in the real line) rather than continuous intervals.

Discrete \( q \)-Hermite polynomials are a family of orthogonal polynomials that arise in the context of the theory of \( q \)-special functions and quantum calculus. They represent a \( q \)-analog of the classical Hermite polynomials, which are well-known in the study of orthogonal polynomials.

Dual Hahn polynomials are a class of orthogonal polynomials that arise in the context of approximation theory, special functions, and mathematical physics. They are part of a broader family of hypergeometric orthogonal polynomials and can be viewed as the dual version of Hahn polynomials.

Dual \( q \)-Hahn polynomials are a class of orthogonal polynomials that arise in the context of basic hypergeometric series and \( q \)-analysis. They can be considered a $q$-analogue of classical orthogonal polynomials, such as the Hahn polynomials.

Favard's theorem is a result in functional analysis and measure theory concerning the Fourier transforms of functions in certain spaces. Specifically, it deals with the conditions under which the Fourier transform of a function in \( L^1 \) space can be represented as a limit of averages of the values of the function.

Gegenbauer polynomials, denoted as \( C_n^{(\lambda)}(x) \), are a family of orthogonal polynomials that generalize Legendre polynomials and Chebyshev polynomials. They arise in various areas of mathematics and are particularly useful in solving problems involving spherical harmonics and certain types of differential equations.

Hahn polynomials are a class of orthogonal polynomials that arise in the context of the theory of orthogonal polynomials on discrete sets. They are named after the mathematician Wolfgang Hahn, who introduced them in the early 20th century. Hahn polynomials are defined for a discrete variable and are often associated with certain types of hypergeometric functions.

The Hall–Littlewood polynomials are a family of symmetric polynomials that play a significant role in various areas of combinatorics, representation theory, and algebraic geometry. They were introduced by Philip Hall and D. E. Littlewood in the mid-20th century as a generalization of the Schur polynomials.

Heckman-Opdam polynomials are a family of orthogonal polynomials that arise in the context of root systems and are closely related to theories in mathematical physics, representation theory, and algebraic combinatorics. They are named after two mathematicians, W. Heckman and E. Opdam, who introduced and studied these polynomials in the context of harmonic analysis on symmetric spaces.

The "Jack function" (also known as the Jack polynomial) is a type of symmetric polynomial that generalizes the Schur polynomials. Jack polynomials depend on a parameter \( \alpha \) and are indexed by partitions. They can be used in various areas of mathematics, including combinatorics, representation theory, and algebraic geometry.

Jacobi polynomials are a class of orthogonal polynomials that arise in various areas of mathematics, including approximation theory, numerical analysis, and the theory of special functions. They are named after the mathematician Carl Gustav Jacob Jacobi.

Koornwinder polynomials are a class of orthogonal polynomials that generalize the basic hypergeometric orthogonal polynomials. They are associated with the root system of type \(C_n\) and are connected to various areas in mathematics, including special functions, combinatorics, and representation theory. The Koornwinder polynomials can be defined using a particular q-orthogonality relation and are characterized by parameters that provide additional flexibility compared to the classical orthogonal polynomials.

Kravchuk polynomials are a class of orthogonal polynomials that arise in the context of combinatorics and probability theory, particularly in relation to the binomial distribution. They are named after the Ukrainian mathematician Kostiantyn Kravchuk.

Little \( q \)-Jacobi polynomials are a family of orthogonal polynomials that arise in the context of q-series and are a particular case of the more general \( q \)-orthogonal polynomials. These polynomials are defined in terms of certain parameters and a variable \( x \), with \( q \) serving as a base for the polynomial’s q-analogue.

Little \( q \)-Laguerre polynomials are a family of orthogonal polynomials that arise in the context of \( q \)-calculus, which is a generalization of classical calculus. They are particularly important in various areas of mathematics and mathematical physics, including combinatorics, special functions, and representation theory.

Macdonald polynomials are a family of symmetric polynomials that arise in the study of algebraic combinatorics, representation theory, and the theory of special functions. They are named after I.G. Macdonald, who introduced them in the context of a generalization of Hall-Littlewood polynomials.

The Mehler kernel is a function that arises in the context of orthogonal polynomials, particularly in relation to the theory of Hermite polynomials and the heat equation. It plays a significant role in probability theory, mathematical physics, and the study of stochastic processes.

The Mehler–Heine formula is a mathematical result concerning orthogonal polynomials and their associated functions. Specifically, it provides a connection between the values of a certain function, defined in terms of orthogonal polynomials, at specific points and their integral representation. More formally, the Mehler–Heine formula typically relates to the context of generating functions for orthogonal polynomials.

Meixner polynomials are a class of orthogonal polynomials that arise in the context of probability theory and various applications in mathematical physics. They are associated with the Meixner distribution, which is a natural generalization of the Poisson distribution and is used to model various types of counting processes.

The Meixner–Pollaczek polynomials are a class of orthogonal polynomials that arise in various areas of mathematics, particularly in spectral theory, probability, and mathematical physics. They can be defined as a part of the broader family of Meixner polynomials, which are associated with certain types of stochastic processes, especially those arising in the context of random walks and queuing theory.

Multiple orthogonal polynomials are a generalization of classical orthogonal polynomials, where the concept of orthogonality is extended to sequences of polynomials with respect to multiple weight functions. This area of research typically arises in contexts where one is dealing with multidimensional problems or when one wants to consider a system of orthogonal polynomials that are related to several different inner products.

Orthogonal polynomials on the unit circle are a class of polynomials that are orthogonal with respect to a specific inner product defined on the unit circle in the complex plane. These polynomials have important applications in various fields, including approximation theory, numerical analysis, and spectral theory.

Plancherel–Rotach asymptotics refers to a set of results in the asymptotic analysis of certain special functions and combinatorial quantities, particularly associated with orthogonal polynomials and probability distributions. The results originally emerged from studying the asymptotic behavior of the zeros of orthogonal polynomials, and they have applications in various areas, including statistical mechanics, random matrix theory, and combinatorial enumeration.

Pseudo-Zernike polynomials are a set of orthogonal polynomials that extend the concept of Zernike polynomials, which are widely used in optics and wavefront analysis. Zernike polynomials form a complete orthogonal basis over the unit disk, which makes them useful for representing wavefronts in applications like optical aberration measurement and correction.

Pseudo-Jacobi polynomials are a class of orthogonal polynomials that are related to the Jacobi polynomials but have some distinct characteristics or domains of applicability. The term "pseudo" typically refers to modifications or generalizations of well-known polynomial families that maintain certain properties or introduce new variables.

Q-Bessel polynomials, also known as Bessel polynomials of the first kind, are specific types of orthogonal polynomials that are related to Bessel functions. These polynomials arise in various areas of mathematics and applied sciences, particularly in solutions to differential equations, mathematical physics, and numerical analysis. Q-Bessel polynomials can be defined through their generating function or through a recurrence relation.

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-Meixner polynomials are a class of orthogonal polynomials that generalize the classical Meixner polynomials. They are typically associated with specific probability distributions, particularly in the context of q-calculus, which is a branch of mathematics dealing with q-series and q-orthogonal polynomials. Meixner polynomials arise in probability theory, especially in relation to certain types of random walks and discrete distributions.

The Q-Meixner–Pollaczek polynomials are a family of orthogonal polynomials that arise in the context of certain special functions and quantum mechanics. They are a generalization of both the Meixner and Pollaczek polynomials and are associated with q-analogues, which are modifications of classic mathematical structures that depend on a parameter \( q \).

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.

Rodrigues' formula is a mathematical expression used to compute powers of rotation matrices in three-dimensional space and to describe the rotation of vectors. It connects the angle of rotation, the axis of rotation, and the vector being rotated.

Rogers polynomials are a family of orthogonal polynomials that arise in the context of approximation theory and special functions. They are closely related to the theory of orthogonal polynomials on the unit circle and have connections to various areas of mathematics, including combinatorics and number theory.

Sobolev orthogonal polynomials are a generalization of classical orthogonal polynomials that arise in the context of Sobolev spaces. In classical approximation theory, orthogonal polynomials, such as Legendre, Hermite, and Laguerre polynomials, are orthogonal with respect to a weight function over a given interval or domain. Sobolev orthogonal polynomials extend this concept by introducing a notion of orthogonality that involves both a weight function and derivatives.

The Stieltjes-Wigert polynomials are a family of orthogonal polynomials that arise in the context of positive definite measures and are associated with a specific weight function on the real line. They are named after mathematicians Thomas Joannes Stieltjes and Hugo Wigert. The Stieltjes-Wigert polynomials can be characterized by the following features: 1. **Orthogonality**: These polynomials are orthogonal with respect to a certain weighted inner product.

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.

Zernike polynomials are a set of orthogonal polynomials defined over a unit disk, which are commonly used in various fields such as optics, imaging science, and surface metrology. They are particularly useful for describing wavefronts and optical aberrations, as they provide a convenient mathematical framework for representing complex shapes and patterns.

Polynomial functions are mathematical expressions that involve sums of powers of variables multiplied by coefficients. A polynomial function in one variable \( x \) can be expressed in the general form: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where: - \( n \) is a non-negative integer representing the degree of the polynomial.

A constant function is a type of mathematical function that always returns the same value regardless of the input. In simpler terms, no matter what value you substitute into a constant function, the output will never change; it will always be a fixed value. Mathematically, a constant function can be expressed in the form: \[ f(x) = c \] where \( c \) is a constant (a specific number) and \( x \) represents the input variable.

A cubic function is a type of polynomial function of degree three, which means that the highest power of the variable (usually denoted as \(x\)) is three.

A linear function is a mathematical function that describes a relationship between two variables that can be graphically represented as a straight line.

A linear function is a type of mathematical function that represents a straight line when graphed on a coordinate plane. In calculus, as well as in algebra, linear functions are defined by the equation of the form: \[ f(x) = mx + b \] Here: - \( f(x) \) is the value of the function at \( x \). - \( m \) is the slope of the line, which indicates how steep the line is.

The Newton polytope is a geometric object associated with a polynomial function, particularly in the context of algebraic geometry and combinatorial geometry. It provides a way to study the roots of a polynomial and the properties of the polynomial itself by examining the combinatorial structure of its coefficients.

A quadratic function is a type of polynomial function of the form: \[ f(x) = ax^2 + bx + c \] where: - \( a \), \( b \), and \( c \) are constants (with \( a \neq 0 \)), - \( x \) is the variable, - \( a \) determines the direction of the parabola (if \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it

A quartic function is a polynomial function of degree four. It can be expressed in the general form: \[ f(x) = ax^4 + bx^3 + cx^2 + dx + e \] where: - \( a, b, c, d, e \) are constants (with \( a \neq 0 \) to ensure that the polynomial is indeed of degree four), - \( x \) is the variable.

A quintic function is a type of polynomial function of degree five. In general, a polynomial function of degree \( n \) can be written in the form: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] For a quintic function, \( n = 5 \).

In mathematics, particularly in algebra, the "ring of polynomial functions" refers to a specific kind of mathematical structure that consists of polynomial functions, along with the operations of addition and multiplication.

Polynomial factorization algorithms are computational methods used to express a polynomial as a product of simpler polynomials, typically of lower degree. These algorithms are important in various fields of mathematics, computer science, and engineering, particularly in areas such as algebra, numerical analysis, control theory, and cryptography. Here are some commonly known algorithms and methods for polynomial factorization: 1. **Factor by Grouping**: This method involves rearranging and grouping terms in the polynomial in order to factor by common factors.

Rational functions are mathematical expressions formed by the ratio of two polynomials. In more formal terms, a rational function \( R(x) \) can be expressed as: \[ R(x) = \frac{P(x)}{Q(x)} \] where \( P(x) \) and \( Q(x) \) are polynomial functions, and \( Q(x) \neq 0 \) (the denominator cannot be zero).

Partial fractions is a mathematical technique used to decompose a rational function into a sum of simpler fractions, called partial fractions. This method is particularly useful in algebra, calculus, and differential equations, as it simplifies the process of integrating rational functions. A rational function is typically expressed as the ratio of two polynomials, say \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

Chebyshev rational functions are specific types of rational functions that are associated with Chebyshev polynomials, which are a sequence of orthogonal polynomials that arise in various areas of numerical analysis, approximation theory, and many applications in engineering and mathematics.

Elliptic rational functions are mathematical functions that arise in the study of elliptic curves and, more generally, in the theory of elliptic functions. They can be thought of as generalizations of rational functions that incorporate properties of elliptic functions. To understand elliptic rational functions, it's helpful to break down the components of the term: 1. **Elliptic Functions:** These are meromorphic functions that are periodic in two directions (often associated with the complex plane's lattice structure).

The Hartogs–Rosenthal theorem is a result in the field of functional analysis, particularly dealing with Banach spaces. It describes a certain property of bounded linear operators between infinite-dimensional Banach spaces.

Legendre rational functions are a family of rational functions constructed from Legendre polynomials, which are orthogonal polynomials defined on the interval \([-1, 1]\). These functions are used in various areas of mathematics, including numerical analysis and approximation theory.