Q-analog
Encyclopedia
Roughly speaking, in mathematics
, specifically in the areas of combinatorics
and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit
as q → 1. Typically, mathematicians are interested in q-analogues that arise naturally, rather than in arbitrarily contriving q-analogues of known results. The earliest q-analog studied in detail is the basic hypergeometric series
, which was introduced in the 19th century.
q-analogs find applications in a number of areas, including the study of fractal
s and multi-fractal measures, and expressions for the entropy
of chaotic dynamical system
s. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian group
s in general (see, for example Indra's pearls
and the Apollonian gasket
) and the modular group
in particular. The connection passes through hyperbolic geometry
and ergodic theory
, where the elliptic integral
s and modular form
s play a prominent role; the q-series themselves are closely related to elliptic integrals.
q-analogs also appear in the study of quantum group
s and in q-deformed superalgebra
s. The connection here is similar, in that much of string theory
is set in the language of Riemann surface
s, resulting in connections to elliptic curve
s, which in turn relate to q-series.
There are two main groups of q-analogs, the "classical" q-analogs, with beginnings in the work of Leonhard Euler
and extended by F. H. Jackson and others. A second group of q-analogs have been developed by Constantino Tsallis
and others.
suggests that we define the q-analog of n, also known as the q-bracket or q-number of n, to be
By itself, the choice of this particular q-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use [n]q as the q-analog of n, one may define the q-analog of the factorial
, known as the q-factorial, by
This q-analog appears naturally in several contexts. Notably, while n! counts the number of permutation
s of length n, we have that [n]q! counts permutations while keeping track of the number of inversions. That is, if inv(w) denotes the number of inversions of the permutation w and Sn denotes the set of permutations of length n, we have
In particular, one recovers the usual factorial by taking the limit as
.
The q-factorial also has a concise definition in terms of the q-Pochammer symbol, a basic building-block of all q-theories:
.
From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients:
which allows q-addition to be defined:
and subtraction:
The q-exponential is defined as:
Q-trigonometric functions, along with a q-Fourier transform have been defined in this context
. Let q be the number of elements in a finite field
. (The number q is then a power of a prime number
, q = pe, so using the letter q is especially appropriate.) Then the number of k-dimensional subspaces of the n-dimensional vector space over the q-element field equals
Letting q approach 1, we get the binomial coefficient
or in other words, the number of k-element subsets of an n-element set.
Thus, one can regard a finite vector space as a q-generalization of a set, and the subspaces as the q-generalization of the subsets of the set. This has been a fruitful point of view in finding interesting new theorems. For example, there are q-analogs of Sperner's theorem and Ramsey theory
.
Q-addition is commutative, associative, has 0 as the identity element, and for q=1 becomes the usual sum of x and y. By inversion, q-subtraction is defined:
The q-product is defined as:
where
is defined to mean x when 
and 0 when 
. Note that for 
, q-division by zero is allowed. By inversion, q-division is defined as:
Many q-functions may be defined using the above building blocks, such as the Tsallis q-exponential and its inverse, the Tsallis q-logarithm which are defined as:
This can be formalized in the field with one element
, which recovers combinatorics as linear algebra over the field with one element: for example, Weyl group
s are simple algebraic group
s over the field with one element.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in the areas of combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
as q → 1. Typically, mathematicians are interested in q-analogues that arise naturally, rather than in arbitrarily contriving q-analogues of known results. The earliest q-analog studied in detail is the basic hypergeometric series
Basic hypergeometric series
In mathematics, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series....
, which was introduced in the 19th century.
q-analogs find applications in a number of areas, including the study of fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s and multi-fractal measures, and expressions for the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
of chaotic dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian group
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
s in general (see, for example Indra's pearls
Indra's Pearls (book)
Indra's Pearls: The Vision of Felix Klein is a geometry book written by David Mumford, Caroline Series and David Wright, and published by Cambridge University Press in 2002....
and the Apollonian gasket
Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga.-Construction:...
) and the modular group
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
in particular. The connection passes through hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
and ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, where the elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...
s and modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s play a prominent role; the q-series themselves are closely related to elliptic integrals.
q-analogs also appear in the study of quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
s and in q-deformed superalgebra
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....
s. The connection here is similar, in that much of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
is set in the language of Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s, resulting in connections to elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s, which in turn relate to q-series.
There are two main groups of q-analogs, the "classical" q-analogs, with beginnings in the work of Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
and extended by F. H. Jackson and others. A second group of q-analogs have been developed by Constantino Tsallis
Constantino Tsallis
Constantino Tsallis is a naturalized Brazilian physicist working in Rio de Janeiro at CBPF, Brazil. He was born in Greece, and grew up in Argentina, where he studied physics at Instituto Balseiro, in Bariloche. In 1974 he received a Doctorat d'Etat et Sciences Physiques degree from the University...
and others.
"Classical"
q-theory Classical q-theory begins with the q-analogs of the nonnegative integers. The equalitysuggests that we define the q-analog of n, also known as the q-bracket or q-number of n, to be
By itself, the choice of this particular q-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use [n]q as the q-analog of n, one may define the q-analog of the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
, known as the q-factorial, by
 |
 |
 |
|
 |
This q-analog appears naturally in several contexts. Notably, while n! counts the number of permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s of length n, we have that [n]q! counts permutations while keeping track of the number of inversions. That is, if inv(w) denotes the number of inversions of the permutation w and Sn denotes the set of permutations of length n, we have
In particular, one recovers the usual factorial by taking the limit as
.The q-factorial also has a concise definition in terms of the q-Pochammer symbol, a basic building-block of all q-theories:
.From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients:
which allows q-addition to be defined:
and subtraction:
The q-exponential is defined as:
Q-trigonometric functions, along with a q-Fourier transform have been defined in this context
Combinatorial q-analogs
The Gaussian coefficients count subspaces of a finite vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. Let q be the number of elements in a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
. (The number q is then a power of a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, q = pe, so using the letter q is especially appropriate.) Then the number of k-dimensional subspaces of the n-dimensional vector space over the q-element field equals
Letting q approach 1, we get the binomial coefficient
or in other words, the number of k-element subsets of an n-element set.
Thus, one can regard a finite vector space as a q-generalization of a set, and the subspaces as the q-generalization of the subsets of the set. This has been a fruitful point of view in finding interesting new theorems. For example, there are q-analogs of Sperner's theorem and Ramsey theory
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
.
Tsallis q-theory
In Tsallis q-theory, one begins by defining the q-addition of two real numbers:Q-addition is commutative, associative, has 0 as the identity element, and for q=1 becomes the usual sum of x and y. By inversion, q-subtraction is defined:
The q-product is defined as:
where
is defined to mean x when and 0 when . Note that for , q-division by zero is allowed. By inversion, q-division is defined as:Many q-functions may be defined using the above building blocks, such as the Tsallis q-exponential and its inverse, the Tsallis q-logarithm which are defined as:
q → 1
Conversely to letting q vary and seeing q-analogs as deformations, one can consider the combinatorial case of q = 1 as a limit of q-analogs as q → 1 (often one cannot simply let q = 1 in the formulae, hence the need to take a limit).This can be formalized in the field with one element
Field with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French-English pun, Fun...
, which recovers combinatorics as linear algebra over the field with one element: for example, Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
s are simple algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s over the field with one element.
See also
- q-derivativeQ-derivativeIn mathematics, in the area of combinatorics, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration-Definition:...
- q-theta function
- Elliptic gamma function
- Stirling numberStirling numberIn mathematics, Stirling numbers arise in a variety of combinatorics problems. They are named after James Stirling, who introduced them in the 18th century. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second...
- Young tableauYoung tableauIn mathematics, a Young tableau is a combinatorial object useful in representation theory. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at...
- Jacobi theta function
- Modular formModular formIn mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
- q-Vandermonde identity
- Field with one elementField with one elementIn mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French-English pun, Fun...
- Gaussian q-distributionGaussian q-distributionIn mathematical physics and probability and statistics, the Gaussian q-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal distribution...
- List of q-analogs