In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product
Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product:
Identities
The finite product can be expressed in terms of the infinite product:
The q-Pochhammer symbol is the subject of a number of q-series identities, particularly the infinite series expansions
Combinatorial interpretation
The q-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of in
We also have that the coefficient of in
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
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 in which the coefficients are functions of q, typically expressions of .[2] Early results are due to Euler, Gauss, and Cauchy. The systematic study begins with Eduard Heine (1843).[3]
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
These numbers are analogues in the sense that
The limit value n! counts permutations of an n-element set S. Equivalently, it counts the number of sequences of nested sets such that contains exactly i elements.[4] 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.[4] 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:[5]
One may further define the q-multinomial coefficients
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, called the q-gamma function, and defined as
See also
- List of q-analogs
- Basic hypergeometric series
- Elliptic gamma function
- Jacobi theta function
- Lambert series
- Pentagonal number theorem
- q-derivative
- q-theta function
- q-Vandermonde identity
- Rogers–Ramanujan identities
- Rogers–Ramanujan continued fraction
References
- ^ Berndt, B. C. "What is a q-series?" (PDF).
- ^ Bruce C. Berndt, What is a q-series?, in Ramanujan Rediscovered: Proceedings of a Conference on Elliptic Functions, Partitions, and q-Series in memory of K. Venkatachaliengar: Bangalore, 1–5 June 2009, N. D. Baruah, B. C. Berndt, S. Cooper, T. Huber, and M. J. Schlosser, eds., Ramanujan Mathematical Society, Mysore, 2010, pp. 31–51.
- ^ Heine, E. "Untersuchungen über die Reihe". J. Reine Angew. Math. 34 (1847), 285–328.
- ^ a b Stanley, Richard P. (2011), Enumerative Combinatorics, vol. 1 (2 ed.), Cambridge University Press, Section 1.10.2.
- ^ Olver; et al. (2010). "Section 17.2". NIST Handbook of Mathematical Functions. p. 421.
- George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
- 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, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
- M.A. Olshanetsky and V.B.K. Rogov (1995), The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions, arXiv:q-alg/9509013.