QBinomial - Maple Help
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QDifferenceEquations

 QPochhammer
 q-Pochhammer symbol
 QBinomial
 q-binomial coefficient
 QBrackets
 q-brackets
 QFactorial
 q-factorial
 QGAMMA
 q-Gamma

 Calling Sequence QPochhammer(a, q, infinity) QPochhammer(a, q, k) QBinomial(n, k, q) QBrackets(k, q) QFactorial(k, q) QGAMMA(a, q)

Parameters

 a - algebraic expression q - name used as the parameter q, or an integer power of a name k - symbolic integer value n - symbolic integer value

Description

 • The QDifferenceEquations package supports five q-hypergeometric terms. They are q-Pochhammer symbol, q-binomial coefficient, q-brackets, q-factorial, and q-Gamma, which correspond to the five functions QPochhammer, QBinomial, QBrackets, QFactorial, and QGAMMA.
 • These functions are placeholders for the q-objects. The command expand allows expansion of these objects. The command $\mathrm{convert}\left(...,\mathrm{QPochhammer}\right)$ allows the re-write of QBinomial, QBrackets, QFactorial, and QGAMMA in terms of QPochhammer symbols.
 • The five q-hypergeometric objects are defined as follows.

$\mathrm{QPochhammer}\left(a,q,\mathrm{\infty }\right)=\prod _{j=0}^{\mathrm{\infty }}\left(1-a{q}^{j}\right)$

$\mathrm{QPochhammer}\left(a,q,k\right)=\left\{\begin{array}{cc}\prod _{j=0}^{k-1}\left(1-a{q}^{j}\right)& 0

 Note that $\mathrm{QPochhammer}\left(\mathrm{seq}\left({a}_{i},i=1..n\right),q,k\right)$ (the compact Gasper and Rahman notation) means ${\prod }_{i=1}^{n}\mathrm{QPochhammer}\left({a}_{i},q,k\right)$.

$\mathrm{QBinomial}\left(n,k,q\right)=\frac{\mathrm{QPochhammer}\left(q,q,n\right)}{\mathrm{QPochhammer}\left(q,q,k\right)\mathrm{QPochhammer}\left(q,q,n-k\right)}$

$\mathrm{QBrackets}\left(k,q\right)=\frac{{q}^{k}-1}{q-1}$

$\mathrm{QFactorial}\left(k,q\right)=\frac{\mathrm{QPochhammer}\left(q,q,k\right)}{{\left(1-q\right)}^{k}}$

$\mathrm{QGAMMA}\left(z,q\right)=\frac{\mathrm{QPochhammer}\left(q,q,\mathrm{\infty }\right){\left(1-q\right)}^{1-z}}{\mathrm{QPochhammer}\left({q}^{z},q,\mathrm{\infty }\right)}$

 • The commands QSimpComb and QSimplify are for simplification of expressions involving these q-objects.
 • This implementation is mainly based on the implementation by H. Boeing, W. Koepf. See the References section.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $\mathrm{expand}\left(\mathrm{QPochhammer}\left(a,q,4\right)\right)$
 $\left({1}{-}{a}\right){}\left({-}{a}{}{q}{+}{1}\right){}\left({-}{a}{}{{q}}^{{2}}{+}{1}\right){}\left({-}{a}{}{{q}}^{{3}}{+}{1}\right)$ (1)
 > $\mathrm{expand}\left(\mathrm{QPochhammer}\left(a,q,-4\right)\right)$
 $\frac{{1}}{\left({1}{-}\frac{{a}}{{{q}}^{{4}}}\right){}\left({1}{-}\frac{{a}}{{{q}}^{{3}}}\right){}\left({1}{-}\frac{{a}}{{{q}}^{{2}}}\right){}\left({1}{-}\frac{{a}}{{q}}\right)}$ (2)
 > $\mathrm{expand}\left(\mathrm{QBrackets}\left(k,q\right)\right)$
 $\frac{{{q}}^{{k}}{-}{1}}{{q}{-}{1}}$ (3)
 > $\mathrm{convert}\left(\mathrm{QBinomial}\left(n,k,q\right),'\mathrm{QPochhammer}'\right)$
 $\frac{{\mathrm{QPochhammer}}{}\left({q}{,}{q}{,}{n}\right)}{{\mathrm{QPochhammer}}{}\left({q}{,}{q}{,}{k}\right){}{\mathrm{QPochhammer}}{}\left({q}{,}{q}{,}{n}{-}{k}\right)}$ (4)
 > $\mathrm{convert}\left(\mathrm{QGAMMA}\left(z,q\right),'\mathrm{QPochhammer}'\right)$
 $\frac{{\mathrm{QPochhammer}}{}\left({q}{,}{q}{,}{\mathrm{\infty }}\right){}{\left({1}{-}{q}\right)}^{{1}{-}{z}}}{{\mathrm{QPochhammer}}{}\left({{q}}^{{z}}{,}{q}{,}{\mathrm{\infty }}\right)}$ (5)
 > $\mathrm{convert}\left(\mathrm{QFactorial}\left(k,q\right),'\mathrm{QPochhammer}'\right)$
 $\frac{{\mathrm{QPochhammer}}{}\left({q}{,}{q}{,}{k}\right)}{{\left({1}{-}{q}\right)}^{{k}}}$ (6)
 > $H≔\frac{{\left(\frac{{\left({q}^{2}-1\right)}^{2}}{{q}^{6}}\right)}^{n}\mathrm{QPochhammer}\left(\frac{1}{-{q}^{5}+{q}^{3}},q,n\right)\mathrm{QPochhammer}\left(\frac{1}{-{q}^{4}+{q}^{2}},q,n\right)\mathrm{QPochhammer}\left(-\frac{1{q}^{3}}{{q}^{2}-1},q,n\right)\mathrm{QPochhammer}\left(-\frac{1}{{q}^{2}},q,n\right)\mathrm{QPochhammer}\left(-\frac{1{q}^{12}}{{q}^{2}-1},q,n\right)\mathrm{QPochhammer}\left(-1,q,n\right)}{\mathrm{QPochhammer}\left(-\frac{1{q}^{2}}{{q}^{2}-1},q,n\right)\mathrm{QPochhammer}\left(-\frac{1}{{q}^{5}},q,n\right){\mathrm{QPochhammer}\left(-\frac{1}{{q}^{4}},q,n\right)}^{2}\mathrm{QPochhammer}\left(-{q}^{4},q,n\right)\mathrm{QPochhammer}\left(\frac{1}{-{q}^{2}+1},q,n\right)}$
 ${H}{≔}\frac{{\left(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{5}}{+}{{q}}^{{3}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{4}}{+}{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{3}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{12}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-1}{,}{q}{,}{n}\right)}{{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{2}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right){}{{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{4}}}{,}{q}{,}{n}\right)}^{{2}}{}{\mathrm{QPochhammer}}{}\left({-}{{q}}^{{4}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{2}}{+}{1}}{,}{q}{,}{n}\right)}$ (7)

Compute the certificate of H (which is a rational function in ${q}^{n}$):

 > $\mathrm{QSimpComb}\left(\frac{\mathrm{subs}\left(n=n+1,H\right)}{H}\right)$
 $\frac{\left({{q}}^{{5}}{-}{{q}}^{{3}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{2}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{n}}{}{{q}}^{{12}}{+}{{q}}^{{2}}{-}{1}\right){}\left({1}{+}{{q}}^{{n}}\right){}\left({{q}}^{{n}}{}{{q}}^{{3}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{{q}}^{{n}}\right)}{\left({{q}}^{{n}}{}{{q}}^{{2}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{2}}{+}{{q}}^{{n}}{-}{1}\right){}{\left({{q}}^{{4}}{+}{{q}}^{{n}}\right)}^{{2}}{}\left({1}{+}{{q}}^{{n}}{}{{q}}^{{4}}\right){}\left({{q}}^{{5}}{+}{{q}}^{{n}}\right)}$ (8)

References

 Boeing, H., and Koepf, W. "Algorithms for q-hypergeometric summation in computer algebra." Journal of Symbolic Computation. Vol. 11. (1999): 1-23.

 See Also