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QDifferenceEquations

 IsQHypergeometricTerm
 test if a given expression is a q-hypergeometric term

 Calling Sequence IsQHypergeometricTerm(H, n, q^n=N, R)

Parameters

 H - function of q^n, algebraic expression n - variable q - name used as the parameter q, usually q N - name R - (optional) name; assigned the computed certificate

Description

 • The IsQHypergeometricTerm(H,n,q^n=N,R) command returns true if $H$ is a q-hypergeometric term of q^n. Otherwise, it returns false.
 A function H is q-hypergeometric of q^n if $\frac{H\left({q}^{n+1}\right)}{H\left({q}^{n}\right)}=R\left({q}^{n}\right)$, a rational function of q^n. $R\left({q}^{n}\right)$ is the certificate of $H\left({q}^{n}\right)$. If the fourth optional argument is included, it is assigned the certificate $R\left(N\right)=R\left({q}^{n}\right)$.
 • This implementation is mainly based on the implementation by H. Boeing and W. Koepf. See the References section.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $T≔\frac{\left({q}^{n}-1\right)\mathrm{QPochhammer}\left(a,q,n\right)\left({\prod }_{k=0}^{n-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({q}^{4}{q}^{k}+1\right)\right)}{{\left({q}^{n}\right)}^{2}}$
 ${T}{≔}\frac{\left({{q}}^{{n}}{-}{1}\right){}{\mathrm{QDifferenceEquations:-QPochhammer}}{}\left({a}{,}{q}{,}{n}\right){}\left({\prod }_{{k}{=}{0}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({{q}}^{{4}}{}{{q}}^{{k}}{+}{1}\right)\right)}{{\left({{q}}^{{n}}\right)}^{{2}}}$ (1)
 > $\mathrm{IsQHypergeometricTerm}\left(T,n,{q}^{n}=N,'R'\right)$
 ${\mathrm{true}}$ (2)
 > $R$
 ${-}\frac{\left({N}{}{{q}}^{{4}}{+}{1}\right){}\left({N}{}{q}{-}{1}\right){}\left({N}{}{a}{-}{1}\right)}{{{q}}^{{2}}{}\left({N}{-}{1}\right)}$ (3)
 > $T≔\mathrm{QBrackets}\left(\frac{n}{2},q\right)$
 ${T}{≔}{\mathrm{QDifferenceEquations:-QBrackets}}{}\left(\frac{{1}}{{2}}{}{n}{,}{q}\right)$ (4)
 > $\mathrm{IsQHypergeometricTerm}\left(T,n,{q}^{n}=N\right)$
 ${\mathrm{false}}$ (5)

References

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