Q-analogs

Q-analogs are generalizations of classical mathematical objects that involve a parameter \( q \). They appear in various branches of mathematics, including algebra, combinatorics, and representation theory. The introduction of the parameter \( q \) typically introduces new structures that retain some properties of the original objects while exhibiting different behaviors.

A Bailey pair is a specific concept in the context of combinatorial identities and combinatorial number theory. It is related to the theory of basic hypergeometric series, which are a generalization of classical hypergeometric series. In particular, a Bailey pair consists of two sequences of numbers, usually denoted as \( (a_n) \) and \( (b_n) \), that satisfy certain combinatorial conditions and can be used to derive identities involving sums or series.

The basic hypergeometric series, also known as the \( q \)-hypergeometric series, is a generalization of the classical hypergeometric series. It involves parameters and is particularly important in various areas of mathematics, including combinatorics, number theory, and q-series.

The Euler function, often denoted as \(\phi(n)\), is also known as Euler's totient function. It counts the number of positive integers up to \(n\) that are coprime to \(n\). Two integers \(a\) and \(b\) are said to be coprime (or relatively prime) if their greatest common divisor (gcd) is 1. The function has both theoretical and practical applications in number theory and cryptography.

F. H. Jackson could refer to different things, depending on the context. One of the more notable mentions could be Frederick Hamilton Jackson, a British geographer and historian, known for his work in the early 20th century. He is recognized for his contributions to the study of geography in relation to human society. If you had a different F. H.

The Gaussian \( q \)-distribution is a generalization of the traditional Gaussian (normal) distribution that incorporates the concept of non-extensivity and is part of the broader family of distributions used in q-statistics (also known as Tsallis statistics). This distribution arises in contexts where systems exhibit long-range correlations and is used to describe phenomena that cannot be accurately characterized by the standard Gaussian distribution due to the presence of heavy tails or other non-standard features.

The Hahn-Exton \( q \)-Bessel function is a special function that generalizes the classical Bessel functions in the context of \( q \)-calculus, which is a mathematical framework that extends traditional calculus to include \( q \)-analogues of various concepts. The \( q \)-Bessel functions arise in various areas of mathematics and theoretical physics, including combinatorics, quantum mechanics, and the theory of orthogonal polynomials.

The Jackson integral is a generalization of the Riemann integral and is associated with the theory of q-calculus. It is named after the mathematician George Johnstone Stokes Jackson, who introduced the concept in the context of q-series and q-analogs of various mathematical concepts.

The Jackson \( q \)-Bessel function is a generalization of the ordinary Bessel function based on \( q \)-calculus, a branch of mathematics that deals with the study of \( q \)-series and \( q \)-difference equations. The concept of \( q \)-Bessel functions arises in the context of quantum calculus and has applications in various areas such as combinatorial mathematics, number theory, and mathematical physics.

A Lambert series is a type of mathematical series named after the mathematician Johann Heinrich Lambert. It is defined in a particular form, usually involving a power series with specific coefficients. The general form of a Lambert series can be expressed as: \[ \sum_{n=1}^{\infty} \frac{n q^n}{1 - q^n} \] where \( |q| < 1 \) is a complex variable.

q-analogs are a generalization of mathematical objects that arise in various areas of mathematics, particularly in combinatorics, number theory, and algebra. They typically involve a parameter \( q \) which, when set to 1, recovers the classical version of the concept.

Mock modular forms are a type of mathematical function that generalize the concept of modular forms. They arise in number theory and have connections to various areas, including combinatorics, topology, and mathematical physics. ### Background A **modular form** is a complex function that satisfies certain transformation properties under the action of a subgroup of the modular group, along with specific growth conditions at infinity.

The \( Q \)-Pochhammer symbol, often denoted as \( (a;q)_n \), is a notation used in the area of q-series and q-special functions. It is a generalization of the standard Pochhammer symbol.

The Q-derivative, also known as the fractional derivative or the q-derivative, is a generalization of the traditional derivative that arises in the context of q-calculus, which is an area of mathematics that extends ideas of calculus, particularly in relation to series and special functions.

The term "Q-exponential" typically refers to a generalization of the standard exponential function in the context of non-extensive statistical mechanics and is associated with the concept of Tsallis entropy. In Tsallis statistics, the Q-exponential function is used to describe systems that exhibit non-extensive behavior, meaning they do not obey the standard additive properties of probability, which are used in classical statistical mechanics.

The quantum dilogarithm is a function that emerges in the context of quantum groups and various areas of mathematical physics, particularly in the study of quantum integrable systems and representation theory. It can be viewed as a noncommutative analog of the classical dilogarithm function.

The Ramanujan theta function, denoted as \(\theta(q)\), is a special function that arises in partition theory and modular forms, and has connections to various areas of mathematics, including combinatorial identities and number theory. It is specifically defined for a complex number \(q\) where \( |q| < 1\).