functions (Question \[q.original\] succeeds if and only if the hypergeometric function $\sum_n u(n;\vec{\alpha},\vec{\beta}) z^n$ is algebraic), Bober [@bober1] was able to fully classify parameters providing a positive answer to Question \[q.original\], if the height $L-K$ is assumed to be 1. For completeness, we now present this classification.

\[thm.bober\] Let $L=K+1$. Assume $\alpha_k \ne \beta_l$ for all $k,l$, that $\sum_k \alpha_k = \sum_l \beta_l$ and that $\gcd(\alpha_1,\cdots,\alpha_k,\beta_1,\cdots,\beta_l)=1.$ Then, $u(n;\vec{\alpha},\vec{\beta})$ is an integer for all $n$ if and only if

1.  $(\vec{\alpha},\vec{\beta})$ takes one of the three forms: $((x+y),(x,y))$ for $gcd(x,y)=1$, $((2x,y),(x,2y,x-y))$ for $gcd(x,y)=1$ and $x>y$ or $((2x,2y),(x,y,x+y))$ for $gcd(x,y)=1$.

2.  $(\vec{\alpha},\vec{\beta})$ is one of 52 sporadic parameter sets, for instance $((30,1),(15,10,6))$, which corresponds under $\Phi$ to $(\vec{\gamma},\vec{\delta}) = ((30,1),(2,3,5))$.

Integral hook ratios {#sec.height1hook}
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Since the conditions presented in Question \[q.hook\] are stronger than in Question \[q.alt\], we expect a subset of the (image under $\Phi$ of the) parameter list presented in Theorem \[thm.bober\] to provide a positive answer to Question \[q.hook\]. In fact, we can show that this stronger condition slims down entirely the infinite families and the sporadic cases, with the exception of one lone set of parameters. This is Theorem \[thm.height1hook\], which we now prove.

We start by 