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