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.

$\mathrm{with}\left(\mathrm{QDifferenceEquations}\right)\:$

$T≔\frac{q^n-1}{{\left(q^n\right)}^2}\mathrm{QPochhammer}\left(a\,q\,n\right)\left(\mathrm{Product}\left(q^4{q}^k+1\,k=0..n-1\right)\right)$

$T≔\frac{\left(q^n-1\right)\mathrm{QPochhammer}\left(a\,q\,n\right)\left(\prod _{k=0}^{n-1}\left(q^4{q}^k+1\right)\right)}{{\left(q^n\right)}^2}$

$\mathrm{IsQHypergeometricTerm}\left(T\,n\,q^n=N\,'R'\right)$

$\mathrm{true}$

$R$

$-\frac{\left(q^{4}N+1\right)\left(Nq-1\right)\left(aN-1\right)}{q^2\left(N-1\right)}$

$T≔\mathrm{QBrackets}\left(\frac{n}{2}\,q\right)$

$T≔\mathrm{QBrackets}\left(\frac{n}{2}\,q\right)$

$\mathrm{IsQHypergeometricTerm}\left(T\,n\,q^n=N\right)$

$\mathrm{false}$

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