Polynomial stubs

The term "polynomial stubs" is not widely recognized in mathematical literature, but it might refer to a few different concepts depending on the context. Below are a couple of possible interpretations: 1. **Polynomial Functions in Partial Fractions:** In the context of calculus or algebra, a "stub" could refer to a part of an expression that needs to be simplified or further investigated, particularly in the context of breaking down a polynomial into partial fractions.

Abel polynomials, named after the mathematician Niels Henrik Abel, are a specific class of polynomials that typically arise in the context of algebra and number theory.

Actuarial polynomials are specific mathematical tools used primarily in actuarial science, often in the context of modeling and calculating insurance liabilities, annuities, and life contingencies. They can be used to represent functions that describe various actuarial processes or outcomes.

Aitken interpolation, also known as Aitken's delta-squared process, is a method used in numerical analysis to improve the convergence of a sequence of approximations to a limit, particularly when working with interpolation polynomials. The primary idea of Aitken interpolation is to accelerate the convergence of a sequence generated by an interpolation process.

The Al-Salam–Ismail polynomials, often denoted \( p_n(x; a, b) \), are a family of orthogonal polynomials that are generalized and belong to the class of basic hypergeometric polynomials. They are named after the mathematicians Al-Salam and Ismail, who introduced them in the context of approximation theory and special functions.

Askey-Wilson polynomials are a family of orthogonal polynomials that play a significant role in the theory of special functions, combinatorics, and mathematical physics. They are a part of the Askey scheme of hypergeometric orthogonal polynomials, which classifies various families of orthogonal polynomials and their relationships.

Bender-Dunne polynomials are a family of orthogonal polynomials that arise in the context of quantum mechanics and mathematical physics. They were introduced by the physicists Carl M. Bender and Peter D. Dunne in their study of non-Hermitian quantum mechanics, which has applications in various fields, including quantum field theory and statistical mechanics. The Bender-Dunne polynomials are particularly notable for their properties in relation to the eigenvalues of certain non-Hermitian Hamiltonians.

The Big \( q \)-Jacobi polynomials are a family of orthogonal polynomials that are part of the larger theory of \( q \)-orthogonal polynomials. They are defined in terms of two parameters, often denoted as \( a \) and \( b \), and a third parameter \( q \) which is a real number between 0 and 1.

The Big \( q \)-Legendre polynomials are a generalization of the classical Legendre polynomials, which arise in various areas of mathematics, including orthogonal polynomial theory and special functions. The \( q \)-analog of mathematical concepts replaces conventional operations with ones that are compatible with the \( q \)-calculus, often leading to new insights and applications, particularly in combinatorial contexts, statistical mechanics, and quantum algebra.

Boas–Buck polynomials are a family of orthogonal polynomials that arise in the study of polynomial approximation theory. They are named after mathematicians Harold P. Boas and Larry Buck, who introduced them in the context of approximating functions on the unit disk. These polynomials can be defined using a specific recursion relation, or equivalently, they can be described using their generating functions.

Boolean polynomials are mathematical expressions that consist of variables that take on values from the Boolean domain, typically 0 and 1. In this context, a Boolean polynomial is constructed using binary operations like AND, OR, and NOT, and it can be expressed in terms of addition (which corresponds to the logical OR operation) and multiplication (which corresponds to the logical AND operation).

Brenke–Chihara polynomials are a specific sequence of polynomials that arise in the context of combinatorics and orthogonal polynomials. They are related to various mathematical areas including approximation theory, numerical analysis, and probability theory. These polynomials can be defined recursively and are often characterized by certain orthogonality conditions concerning a weight function over an interval. The exact properties and applications can vary significantly depending on the context in which the polynomials are used.

A caloric polynomial is a mathematical concept arising in the context of potential theory and various applications in mathematics, particularly in the study of harmonic functions. While not as widely known as some other types of polynomials, the term is often associated with the following defining properties: 1. **General Definition**: A caloric polynomial can be understood as a polynomial that satisfies specific boundary conditions related to the heat equation or to the Laplace equation.

The Carlitz-Wan conjecture is a conjecture in number theory related to the distribution of roots of polynomials over finite fields. Specifically, it is concerned with the number of roots of certain families of polynomials in the context of function fields. The conjecture was posed by L. Carlitz and J. Wan and suggests a specific behavior regarding the number of rational points (or roots) of certain algebraic equations over finite fields.

In the context of algebra and algebraic structures, particularly in the theory of rings and algebras, a **central polynomial** typically refers to a polynomial in several variables that commutes with all elements of a certain algebraic structure, such as a matrix algebra or a group algebra.

Charlier polynomials are a sequence of orthogonal polynomials that arise in probability and analysis. They are a specific case of hypergeometric polynomials and can be defined in the context of the Poisson distribution. The Charlier polynomials \( C_n(x; a) \) are defined as follows: \[ C_n(x; a) = \sum_{k=0}^{n} \frac{(-1)^{n-k}}{(n-k)!

Chihara–Ismail polynomials, also known as Chihara polynomials, are a family of orthogonal polynomials that arise in mathematical physics, particularly in the context of quantum mechanics and statistical mechanics. They are typically defined with respect to a specific weight function over an interval, and they are generated by a certain orthogonality condition.

Continuous \( q \)-Legendre polynomials are a family of orthogonal polynomials that extend classical Legendre polynomials into the realm of \( q \)-calculus. They arise in various areas of mathematics and physics, particularly in the study of orthogonal functions, approximation theory, and in the context of quantum groups and \( q \)-series.

Denisyuk polynomials refer to a special class of polynomial curves in the context of algebraic geometry and computer graphics. Specifically, they are named after the Russian mathematician and physicist Mikhail Denisyuk, who made contributions to the field of holography and optical phenomena, including the study of polynomials that describe certain geometric properties.

The dual q-Krawtchouk polynomials are a family of orthogonal polynomials associated with the discrete probability distributions arising from the q-analog of the Krawtchouk polynomials. These polynomials arise in various areas of mathematics and have applications in combinatorics, statistical mechanics, and quantum groups. The Krawtchouk polynomials themselves are defined in terms of binomial coefficients and arise in the study of discrete distributions, particularly with respect to the binomial distribution.

The FGLM algorithm, which stands for "Feldman, Gilg, Lichtenstein, and Maler" algorithm, is primarily a method used in the field of computational intelligence and learning theory, specifically focused on learning finite automata. The FGLM algorithm is designed to infer the structure of a finite automaton from a given set of input-output pairs (also known as labeled sequences).

Faber polynomials are a sequence of orthogonal polynomials that arise in the context of complex analysis and approximation theory. They are particularly associated with the problem of approximating analytic functions on the unit disk in the complex plane. For a given analytic function \( f \) defined on the unit disk, the Faber polynomial \( P_n(z) \) can be used to construct an approximation of \( f \) through a series representation.

A Fekete polynomial is a specific type of polynomial that arises in the context of approximation theory and numerical analysis. It is typically associated with the study of orthogonal polynomials and their properties. Fekete polynomials are named after the Hungarian mathematician A. Fekete. They are used in the context of finding optimal distributions of points, particularly in relation to minimizing the potential energy of point distributions in certain spaces.

Generalized Appell polynomials are a family of orthogonal polynomials that generalize the classical Appell polynomials. Appell polynomials are a set of polynomials \(A_n(x)\) such that the \(n\)-th polynomial can be defined via a generating function or a differential equation relationship. Specifically, Appell polynomials satisfy the condition: \[ A_n'(x) = n A_{n-1}(x) \] with a given initial condition.

Geronimus polynomials are a class of orthogonal polynomials that arise in the context of discrete orthogonal polynomial theory. They are named after the mathematician M. Geronimus, who contributed to the theory of orthogonal polynomials. Geronimus polynomials can be defined as a modification of the classical orthogonal polynomials, such as Hermite, Laguerre, or Jacobi polynomials.

Gottlieb polynomials are a specific sequence of polynomials that arise in various mathematical contexts, particularly in number theory and combinatorics. They are defined through generators related to specific algebraic structures. In the context of special functions, Gottlieb polynomials can be related to matrix theory and possess properties similar to those of classical orthogonal polynomials. The explicit form and properties of these polynomials depend on how they are defined, typically involving combinatorial coefficients or generating functions.

Gould polynomials are a family of orthogonal polynomials that are particularly associated with the study of combinatorial identities and certain types of generating functions. They are often denoted using the notation \(P_n(x)\), where \(n\) is a non-negative integer and \(x\) represents a variable. These polynomials can arise in various mathematical contexts, including approximation theory, numerical analysis, and special functions.

Heine–Stieltjes polynomials are a generalization of classical orthogonal polynomials, named after mathematicians Heinrich Heine and Thomas Joannes Stieltjes. These polynomials arise in the context of continuous fraction expansions and orthogonal polynomial theory.

Hudde's Rules refer to a set of guidelines used in organic chemistry for determining the stability of reaction intermediates, particularly carbocations and carbanions. These rules help predict the relative reactivity and stability of different carbocation species based on their structure and the substituents attached to them.

Humbert polynomials are a class of orthogonal polynomials that arise in the context of mathematical analysis and number theory. They are named after the mathematician Humbert, who studied various properties of these polynomials. Humbert polynomials can be used in various applications, including approximation theory, numerical analysis, and even in solving certain types of differential equations.

The Kauffman polynomial is an important invariant in knot theory, a branch of mathematics that studies the properties of knots. It was introduced by Louis Kauffman in the 1980s and serves as a polynomial invariant of oriented links in three-dimensional space. The Kauffman polynomial can be defined for a link diagram, which is a planar representation of a link with crossings marked.

The Kharitonov region, also known as Kharitonovsky District, is a federal subject of Russia, located in the Siberian region. However, specific information about the Kharitonov region is limited, as it might refer to a less prominent area or could be a misnomer for a specific district within a larger region that is commonly known by another name.

Konhauser polynomials are a sequence of polynomials that arise in the context of combinatorics and algebraic topology, particularly in the study of certain generating functions and combinatorial structures. They are named after the mathematician David Konhauser. These polynomials can be defined through various combinatorial interpretations and have applications in enumerating certain types of objects, such as trees or partitions.

The term "LLT polynomial" refers to a specific type of polynomial associated with certain combinatorial and algebraic structures. It is named after its developers, Lau, Lin, and Tsiang. LLT polynomials are particularly relevant in the context of symmetric functions and the representation theory of symmetric groups. LLT polynomials can be defined in the setting of generating functions and are often used to study various combinatorial objects, such as partitions and tableaux.

Lommel polynomials are a set of orthogonal polynomials that arise in the context of Bessel functions and have important applications in various areas of mathematical analysis, particularly in problems related to wave propagation, optics, and differential equations.

Mahler polynomials are a family of orthogonal polynomials that arise in the context of number theory and special functions. They are associated with the Mahler measure, which is a concept used to study the growth of certain types of polynomials. The Mahler polynomials can be defined in terms of a generating function or recursively.

Mott polynomials are a class of orthogonal polynomials that play a significant role in various areas of mathematics, particularly in the realm of functional analysis and the theory of orthogonal functions. They are named after the British physicist and mathematician N.F. Mott, who made contributions to the understanding of complex systems.

Narumi polynomials are a class of polynomials used in number theory and combinatorics, particularly in the context of enumerating certain types of combinatorial structures or in the study of generating functions. They are named after the Japanese mathematician Katsura Narumi. The Narumi polynomials can be defined by specific recurrence relations or generating functions, and they often arise in problems related to partitions, compositions, or other combinatorial constructs.

Padovan polynomials are a sequence of polynomials that arise in the study of number theory and combinatorial mathematics.

Peters polynomials are a sequence of orthogonal polynomials associated with the theory of orthogonal functions and are specifically related to the study of function approximation and interpolation. They can be regarded as a specific case of orthogonal polynomials on specific intervals or with certain weights. While "Peters polynomials" might not be as widely referenced as, say, Legendre or Chebyshev polynomials, they represent an interesting area of study within numerical analysis and mathematical approximation.

Pidduck polynomials are a sequence of orthogonal polynomials that arise in the context of serial correlation and some applications in probability theory and statistics. They are named after the mathematician who studied them, Arthur Pidduck. These polynomials can be defined through a recurrence relation or in terms of an explicit formula involving factorials and powers. They typically exhibit certain orthogonality properties with respect to a weight function over a specified interval.

Pincherle polynomials are a class of polynomials that arise in the context of functional analysis and operator theory, particularly in the study of linear differential and difference equations. Named after the Italian mathematician Antonio Pincherle, these polynomials can be defined through certain recurrence relations or orthogonality properties. In a more specific context, Pincherle polynomials can be used to express solutions to certain classes of problems involving linear transformations or series expansions.

A polylogarithmic function is a type of mathematical function that generalizes the logarithm and can be expressed in terms of the logarithm raised to various powers.

The principal root of unity specifically refers to the complex numbers that satisfy the equation \( z^n = 1 \) for a positive integer \( n \). These roots have the form: \[ z_k = e^{2\pi i k / n} \] for \( k = 0, 1, 2, \ldots, n-1 \).

The Q-Konhauser polynomials, also known as the Q-Konhauser sequence, are a family of orthogonal polynomials that arise in certain combinatorial contexts, particularly in the study of enumerative combinatorics and lattice paths. These polynomials can be used to encode distributions or to solve recurrence relations that have combinatorial interpretations.

The term "quasi-polynomial" refers to a type of mathematical function or expression that generalizes the concept of polynomial functions.

Rainville polynomials are a sequence of orthogonal polynomials that arise in the context of asymptotic analysis and approximations in mathematical physics. They are named after the mathematician Edward D. Rainville, who contributed to their study. These polynomials can be associated with certain weight functions in integration, and they often appear in problems related to probability, statistics, and other areas of applied mathematics.

Rogers–Szegő polynomials are a sequence of orthogonal polynomials that arise in the theory of special functions, particularly in the context of approximation theory and the study of orthogonal functions. They are associated with certain weight functions over the unit circle and have applications in various areas including combinatorics, number theory, and mathematical physics. The Rogers–Szegő polynomials can be defined in terms of a generating function.

In mathematics, the term "secondary polynomials" is not a standard term and may not have a specific definition universally recognized across mathematical literature. It might refer to various concepts depending on the context in which it is used.

Sieved Jacobi polynomials are a special class of orthogonal polynomials that are derived from Jacobi polynomials through a sieving process. To understand this concept, we first need to look at Jacobi polynomials themselves.

Sieved Pollaczek polynomials are a class of polynomials that arise in the context of orthogonal polynomials, specifically in relation to the Pollaczek polynomials. The standard Pollaczek polynomials are a type of orthogonal polynomial that have applications in various areas, such as approximation theory, special functions, and mathematical physics.

Sieved orthogonal polynomials are a class of orthogonal polynomials that are defined with respect to a weight function, where the weight function is modified or "sieved" to omit certain values or intervals. This sieving process leads to a new set of polynomials that retain orthogonality properties, but only over a specified subset of points.

Sieved ultraspherical polynomials, more commonly referred to in the context of orthogonal polynomials, are a specific type of polynomial that arises from the study of special functions and approximation theory. To understand them better, it's useful to break down the terms: 1. **Ultraspherical Polynomials**: These are also known as Gegenbauer polynomials.

Sister Celine's polynomials are a special class of polynomials that arise in the context of combinatorics and algebra. They are defined using a recursive relation similar to that of binomial coefficients.

A **sparse polynomial** is a polynomial in which most of the coefficients are zero, meaning that it has a relatively small number of non-zero terms compared to the total possible terms in the polynomial. This sparsity can significantly affect computations involving the polynomial, making certain operations more efficient.

Stieltjes polynomials are a sequence of orthogonal polynomials that arise in the context of Stieltjes moment problems and are closely related to continued fractions, special functions, and various areas of mathematical analysis. In general, Stieltjes polynomials may be defined for a given positive measure on the real line.

The Szegő polynomials are a sequence of orthogonal polynomials that arise in the context of approximating functions on the unit circle and in the study of analytic functions. They are particularly related to the theory of Fourier series and have applications in various areas, including signal processing and control theory. ### Definition The Szegő polynomials can be defined in terms of their generating function or through specific recurrence relations.

Tian yuan shu, or the "Heavenly Element Method," is a traditional Chinese mathematical system that is primarily concerned with solving equations. It is an ancient technique that originated from China's rich mathematical history and was used extensively in dealing with polynomial equations. In tian yuan shu, problems are typically formulated in terms of a single variable, and the solutions are often derived geometrically or through specific numerical methods.

Tricomi–Carlitz polynomials are a class of polynomials that arise in the study of $q$-analogues in the context of basic hypergeometric series and combinatorial identities. They are named after the mathematicians Francesco Tricomi and Leonard Carlitz, who studied these polynomials in relation to $q$-series. These polynomials can be defined through various generating functions and properties related to $q$-binomial coefficients.

A trinomial is a polynomial that consists of three terms. It is typically expressed in the standard form as: \[ ax^2 + bx + c \] where \( a \), \( b \), and \( c \) are constants (real numbers), and \( x \) is the variable. The term "trinomial" is derived from "tri," meaning three, indicating that it has three distinct terms.

A unimodular polynomial matrix is a matrix of polynomials with coefficients in a polynomial ring such that it has a multiplicative inverse that is also a polynomial matrix.

In the context of mathematics, particularly in algebra and modular forms, "Wall polynomials" often refer to certain types of polynomials associated with combinatorial structures, algebraic geometries, or specific number theoretic problems. However, it is possible that you are referring to the Wall polynomials associated with the theory of modular forms and the theory of partitions. Wall polynomials can arise in the study of modular forms, often in relation to congruences and partition identities.

Wilson polynomials, denoted as \( W_n(x) \), are a class of orthogonal polynomials that arise in the context of probability theory and statistical mechanics. They are defined on the interval \( (0, 1) \) and are associated with the Beta distribution. Wilson polynomials can be expressed using the following formula: \[ W_n(x) = \frac{n!}{(n + 1)!