In
mathematical area of
combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
with
It is a
''q''-analog of the
Pochhammer symbol , in the sense that
The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of
basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of
generalized hypergeometric series.
Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product:
This is an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
of ''q'' in the interior of the
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose ...
, and can also be considered as a
formal power series in ''q''. The special case
is known as
Euler's function
In mathematics, the Euler function is given by
:\phi(q)=\prod_^\infty (1-q^k),\quad , q, A000203
On account of the identity \sum_ d = \sum_ \frac, this may also be written as
:\ln(\phi(q)) = -\sum_^\infty \frac \sum_ d.
Also if a,b\in\mathbb^ ...
, and is important in
combinatorics,
number theory, and the theory of
modular forms.
Identities
The finite product can be expressed in terms of the infinite product:
which extends the definition to negative integers ''n''. Thus, for nonnegative ''n'', one has
and
Alternatively,
which is useful for some of the generating functions of partition functions.
The ''q''-Pochhammer symbol is the subject of a number of ''q''-series identities, particularly the infinite series expansions
and
which are both special cases of the
''q''-binomial theorem:
Fridrikh Karpelevich found the following identity (see for the proof):
Combinatorial interpretation
The ''q''-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of
in
is the number of partitions of ''m'' into at most ''n'' parts.
Since, by conjugation of partitions, this is the same as the number of partitions of ''m'' into parts of size at most ''n'', by identification of generating series we obtain the identity
as in the above section.
We also have that the coefficient of
in
is the number of partitions of ''m'' into ''n'' or ''n''-1 distinct parts.
By removing a triangular partition with ''n'' − 1 parts from such a partition, we are left with an arbitrary partition with at most ''n'' parts. This gives a weight-preserving bijection between the set of partitions into ''n'' or ''n'' − 1 distinct parts and the set of pairs consisting of a triangular partition having ''n'' − 1 parts and a partition with at most ''n'' parts. By identifying generating series, this leads to the identity
also described in the above section.
The reciprocal of the function
similarly arises as the generating function for the
partition function,
, which is also expanded by the second two
q-series expansions given below:
The
''q''-binomial theorem itself can also be handled by a slightly more involved combinatorial argument of a similar flavor (see also the expansions given in the
next subsection).
Similarly,
Multiple arguments convention
Since identities involving ''q''-Pochhammer symbols so frequently involve products of many symbols, the standard convention is to write a product as a single symbol of multiple arguments:
''q''-series
A ''q''-series is a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
in which the coefficients are functions of ''q'', typically expressions of
. Early results are due to
Euler,
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, and
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
. The systematic study begins with
Eduard Heine (1843).
Relationship to other ''q''-functions
The ''q''-analog of ''n'', also known as the ''q''-bracket or ''q''-number of ''n'', is defined to be
From this one can define the ''q''-analog of the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) ...
, the ''q''-factorial, as
These numbers are analogues in the sense that
and so also
The limit value ''n''! counts
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
s of an ''n''-element set ''S''. Equivalently, it counts the number of sequences of nested sets
such that
contains exactly ''i'' elements.
[, Section 1.10.2.] By comparison, when ''q'' is a prime power and ''V'' is an ''n''-dimensional vector space over the field with ''q'' elements, the ''q''-analogue
is the number of complete flags in ''V'', that is, it is the number of sequences
of subspaces such that
has dimension ''i''.
The preceding considerations suggest that one can regard a sequence of nested sets as a flag over a conjectural
field with one element.
A product of negative integer ''q''-brackets can be expressed in terms of the ''q''-factorial as
From the ''q''-factorials, one can move on to define the ''q''-binomial coefficients, also known as the
Gaussian binomial coefficients, as
where it is easy to see that the triangle of these coefficients is symmetric in the sense that
for all
.
One can check that
One can also see from the previous recurrence relations that the next variants of the
-binomial theorem are expanded in terms of these coefficients as follows:
One may further define the ''q''-multinomial coefficients
where the arguments
are nonnegative integers that satisfy
. The coefficient above counts the number of flags
of subspaces in an ''n''-dimensional vector space over the field with ''q'' elements such that
.
The limit
gives the usual multinomial coefficient
, which counts words in ''n'' different symbols
such that each
appears
times.
One also obtains a ''q''-analog of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...
, called the
q-gamma function
In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by
\Gamma_q(x) = (1-q)^\prod_^\infty \frac=(1-q) ...
, and defined as
This converges to the usual gamma function as ''q'' approaches 1 from inside the unit disc. Note that
for any ''x'' and
for non-negative integer values of ''n''. Alternatively, this may be taken as an extension of the ''q''-factorial function to the real number system.
See also
*
Basic hypergeometric series
*
Elliptic gamma function In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by , and can be expressed in terms of the triple gam ...
*
Jacobi theta function
*
Lambert series
*
Pentagonal number theorem
*
''q''-derivative
*
''q''-theta function
*
''q''-Vandermonde identity
*
Rogers–Ramanujan identities In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Sriniv ...
*
Rogers–Ramanujan continued fraction
References
* George Gasper and
Mizan Rahman, ''Basic Hypergeometric Series, 2nd Edition'', (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. .
* Roelof Koekoek and Rene F. Swarttouw,
The Askey scheme of orthogonal polynomials and its q-analogues', section 0.2.
* Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , ,
*M.A. Olshanetsky and V.B.K. Rogov (1995), The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions, arXiv:q-alg/9509013.
External links
*
*
*
*
* {{MathWorld, urlname=q-BinomialCoefficient, title=''q''-Binomial Coefficient
Number theory
Q-analogs