Q-Pochhammer symbol explained

In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product$$(a;q)\_n\; =\; \backslash prod\_^\; (1-aq^k)=(1-a)(1-aq)(1-aq^2)\backslash cdots(1-aq^),$$with

(a;q) _{0} = 1 . It is a q-analog of the Pochhammer symbol (x) _{n} = x(x +1 ) ... (x + n -1 ) , in the sense that$$\backslash lim\_\; \backslash frac\; =\; (x)\_n.$$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:$$(a;q)\_\backslash infty\; =\; \backslash prod\_^\; (1-aq^k).$$This is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q. The special case$$\backslash phi(q)\; =\; (q;q)\_\backslash infty=\backslash prod\_^\backslash infty\; (1-q^k)$$is known as Euler's function, 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:$$(a;q)\_n\; =\; \backslash frac,$$which extends the definition to negative integers n. Thus, for nonnegative n, one has$$(a;q)\_\; =\; \backslash frac=\backslash prod\_^n\; \backslash frac$$and$$(a;q)\_\; =\; \backslash frac\; .$$Alternatively,$$\backslash prod\_^\backslash infty\; (1-aq^k)=(aq^n;q)\_\backslash infty\; =\; \backslash frac,$$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$$(x;q)\_\backslash infty\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; x^n$$and$$\backslash frac=\backslash sum\_^\backslash infty\; \backslash frac,$$which are both special cases of the q-binomial theorem:$$\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; x^n.$$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

q ^{m} a ^{n} in$$(a;q)\_\backslash infty^\; =\; \backslash prod\_^\; (1-aq^k)^$$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$$(a;q)\_\backslash infty^\; =\; \backslash sum\_^\backslash infty\; \backslash left(\backslash prod\_^k\; \backslash frac\; \backslash right)\; a^k\; =\; \backslash sum\_^\backslash infty\; \backslash frac$$as in the above section.

We also have that the coefficient of

q ^{m} a ^{n} in$$(-a;q)\_\backslash infty\; =\; \backslash prod\_^\; (1+aq^k)$$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$$(-a;q)\_\backslash infty\; =\; \backslash prod\_^\backslash infty\; (1+aq^k)\; =\; \backslash sum\_^\backslash infty\; \backslash left(q^\; \backslash prod\_^k\; \backslash frac\backslash right)\; a^k\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; a^k$$also described in the above section. The reciprocal of the function

(q) _{infty} : = (q; q) _{infty} similarly arises as the generating function for the partition function, p(n) , which is also expanded by the second two q-series expansions given below:^[1] $$\backslash frac\; =\; \backslash sum\_\; p(n)\; q^n\; =\; \backslash sum\_\; \backslash frac\; =\; \backslash sum\_\; \backslash frac.$$

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,$$(q;\; q)\_\; =\; 1\; -\; \backslash sum\_\; q^(q;\; q)\_n\; =\; \backslash sum\_\; q^\backslash frac.$$

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:$$(a\_1,a\_2,\backslash ldots,a\_m;q)\_n\; =\; (a\_1;q)\_n\; (a\_2;q)\_n\; \backslash ldots\; (a\_m;q)\_n.$$

q-series

See main article: q-analog. A q-series is a series in which the coefficients are functions of q, typically expressions of

(a; q) _{n} .^[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$$[n]\_q=\backslash frac.$$From this one can define the q-analog of the factorial, the q-factorial, as

These numbers are analogues in the sense that $$\backslash lim\_[n]\_q\; =\; n,$$and so also $$\backslash lim\_[n]!\_q\; =\; n!.$$

The limit value n! counts permutations of an n-element set S. Equivalently, it counts the number of sequences of nested sets

E _{1} \subset E _{2} \subset … \subset E _{n} = S such that E _{i} 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 [n]! _{q} is the number of complete flags in V, that is, it is the number of sequences V _{1} \subset V _{2} \subset … \subset V _{n} = V of subspaces such that V _{i} 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$$\backslash prod\_^n\; [-k]\_q\; =\; \backslash frac$$

From the q-factorials, one can move on to define the q-binomial coefficients, also known as the Gaussian binomial coefficients, as$$\backslash beginn\backslash \backslash k\backslash end\_q=\backslash frac,$$

where it is easy to see that the triangle of these coefficients is symmetric in the sense that

\begin{bmatrix} n \ m \end{bmatrix} _{q} = \begin{bmatrix} n \ n - m \end{bmatrix} _{q}

for all

0 \leq m \leq n . One can check that

One can also see from the previous recurrence relations that the next variants of the

q -binomial theorem are expanded in terms of these coefficients as follows:^[5]

One may further define the q-multinomial coefficients$$\backslash beginn\backslash \backslash k\_1,\; \backslash ldots,k\_m\backslash end\_q=\backslash frac,$$where the arguments

k _{1,} \ldots, k _{m} are nonnegative integers that satisfy \sum ^{m} _{i =1} k _{i} = n . The coefficient above counts the number of flagsV _{1} \subset ... \subset V _{m} of subspaces in an n-dimensional vector space over the field with q elements such that \dim V _{i} = \sum ^{i} _{j =1} k _{j} .

The limit

q\to 1 gives the usual multinomial coefficient {n\choose k _{1,...} ,k _{m}} , which counts words in n different symbols \{s _{1,...,s} _{m\}} such that each s _{i} appears k _{i} times.

One also obtains a q-analog of the gamma function, called the q-gamma function, and defined as$$\backslash Gamma\_q(x)=\backslash frac$$This converges to the usual gamma function as q approaches 1 from inside the unit disc. Note that$$\backslash Gamma\_q(x+1)=[x]\_q\backslash Gamma\_q(x)$$for any x and$$\backslash Gamma\_q(n+1)=[n]!\_q$$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

• 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

• 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.

Notes and References

1. Web site: Berndt. B. C.. What is a q-series?.
2. 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.
3. Web site: Heine. E.. Untersuchungen über die Reihe. J. Reine Angew. Math. 34 (1847), 285–328.
4. , Section 1.10.2.
5. Book: Olver . et al.. NIST Handbook of Mathematical Functions. 2010. Section 17.2. 421.