In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
In mathematics, the Gaussian or ordinary hypergeometric function2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points (RSPs) to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and . The equation is also known as the Papperitz equation.
In mathematics, basic hypergeometric series, or hypergeometric q-series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base.
In quantum statistical mechanics, the von Neumann entropy, named after John von Neumann, is the extension of classical Gibbs entropy concepts to the field of quantum mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is
In mathematics, a matrix function is a function which maps a matrix to another matrix.
In mathematics, the Askey–Wilson polynomials are a family of orthogonal polynomials introduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. Askey–Wilson polynomials are the special case of Macdonald polynomials for the non-reduced affine root system of type, and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system.
In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilson (1980) that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials.
In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients.
In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Askey & Wilson (1979). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
In mathematics, Dixon's identity is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms.
In mathematics, the E-function was introduced by Thomas Murray MacRobert (1937–1938) to extend the generalized hypergeometric series pFq(·) to the case p > q + 1. The underlying objective was to define a very general function that includes as particular cases the majority of the special functions known until then. However, this function had no great impact on the literature as it can always be expressed in terms of the Meijer G-function, while the opposite is not true, so that the G-function is of a still more general nature.
In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
In mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
In mathematics, the little q-Laguerre polynomialspn(x;a|q) or Wall polynomialsWn(x; b,q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to a continued fraction studied by Wall (1941). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky within the context of probability distribution functions in statistics. They form an orthogonal subset of a more general family of little-known Routh polynomials introduced by Edward John Routh in 1884. The term Romanovski polynomials was put forward by Raposo, with reference to the so-called 'pseudo-Jacobi polynomials in Lesky's classification scheme. It seems more consistent to refer to them as Romanovski–Routh polynomials, by analogy with the terms Romanovski–Bessel and Romanovski–Jacobi used by Lesky for two other sets of orthogonal polynomials.
