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q-Binomial Coefficient

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The q-binomial coefficient is a q-analog for the binomial coefficient, also called a Gaussian coefficient or a Gaussian polynomial. A q-binomial coefficient is given by

[n; m]_q=((q)_n)/((q)_m(q)_(n-m))=product_(i=0)^(m-1)(1-q^(n-i))/(1-q^(i+1)),
(1)

where

(q)_k=product_(m=1)^infty(1-q^m)/(1-q^(k+m))
(2)

is a q-series (Koepf 1998, p. 26). For k,n in N,

[n; k]_q=([n]_q!)/([k]_q![n-k]_q!),
(3)

where [n]_q! is a q-factorial (Koepf 1998, p. 30). The q-binomial coefficient can also be defined in terms of the q-brackets [k]_q by

[n; k]_q={product_(i=1)^(k)([n-i+1]_q)/([i]_q) for 0<=k<=n; 0 otherwise.
(4)

The q-binomial is implemented in the Wolfram Language as QBinomial[n, m, q].

For q->1^-, the q-binomial coefficients turn into the usual binomial coefficient.

The special case

[n]_q=[n; 1]_q=(1-q^n)/(1-q)
(5)

is sometimes known as the q-bracket.

The q-binomial coefficient satisfies the recurrence equation

[n+1; k]_q=q^k[n; k]_q+[n; k-1]_q,
(6)

for all n>=1 and 1<=k<=n, so every q-binomial coefficient is a polynomial in q. The first few q-binomial coefficients are

[2; 1]_q=(1-q^2)/(1-q)=1+q
(7)
[3; 1]_q=[3; 2]_q=(1-q^3)/(1-q)=1+q+q^2
(8)
[4; 1]_q=[4; 3]_q=(1-q^4)/(1-q)=1+q+q^2+q^3
(9)
[4; 2]_q=((1-q^3)(1-q^4))/((1-q)(1-q^2))=1+q+2q^2+q^3+q^4.
(10)

From the definition, it follows that

[n; 1]_q=[n; n-1]_q=sum_(i=0)^(n-1)q^i.
(11)

Additional identities include

([n+1; k+1]_q)/([n; k+1]_q)=(1-q^(n+1))/(1-q^(n-k))
(12)
([n+1; k+1]_q)/([n+1; k]_q)=(1-q^(n-k+1))/(1-q^(k+1)).
(13)

The q-binomial coefficient [n; m]_q can be constructed by building all m-subsets of {1,2,...,n}, summing the elements of each subset, and taking the sum

[n; m]_q=sum_(i)q^(s_i-m(m+1)/2)
(14)

over all subset-sums s_i (Kac and Cheung 2001, p. 19).

qBinomial

The q-binomial coefficient [m+n; m]_q can also be interpreted as a polynomial in q whose coefficient q^k counts the number of distinct partitions of k elements which fit inside an m×n rectangle. For example, the partitions of 1, 2, 3, and 4 are given in the following table.

npartitions
0{}
1{{1}}
2{{2},{1,1}}
3{{3},{2,1},{1,1,1}}
4{{4},{3,1},{2,2},{2,1,1},{1,1,1,1}}

Of these, {}, {1}, {2}, {1,1}, {2,1}, and {2,2} fit inside a 2×2 box. The counts of these having 0, 1, 2, 3, and 4 elements are 1, 1, 2, 1, and 1, so the (4, 2)-binomial coefficient is given by

[4; 2]_q=1+q+2q^2+q^3+q^4,
(15)

as above.

See also

Binomial Coefficient, Cauchy Binomial Theorem, Grid Shading Problem, q-Bracket, q-Series, Stieltjes-Wigert Polynomial

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References

Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.Kac, V. Cheung, P. Quantum Calculus. New York:Springer-Verlag, 2001.Koekoek, R. and Swarttouw, R. F. "The q-Gamma Function and the q-Binomial Coefficient." §0.3 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 10-11, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 26, 1998.

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q-Binomial Coefficient

Cite this as:

Weisstein, Eric W. "q-Binomial Coefficient." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-BinomialCoefficient.html

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