Contributions to the proof theory of hypergeometric identities

Lily Yen, University of Pennsylvania


In 1992 Wilf and Zeilberger introduced the following terminology: A hypergeometric term is a function $F(k\sb1,k\sb2,\...,k\sb{r})$ such that, for all $i\in \{1,2,\...,r\},$ the ratio $${F(k\sb1,\...,k\sb{i-1}, k\sb{i} + 1, k\sb{i + 1},\...,k\sb{r})\over F(k\sb1,\...,k\sb{r})}$$is a rational function in all the variables. They also introduced the rather technical concept of admissible proper-hypergeometric terms; "most interesting" hypergeometric terms are admissible and proper. We prove the following: Given an integer $n\sb0$ and an admissible proper-hypergeometric term $F(n,k),$ there exists a pre-computable integer $n\sb1$ such that if $\sum\sb{k}F(n,k) = 1$ for $n\sb0\le n\le n\sb1,$ then $\sum\sb{k}F(n,k) = 1$ for $all\ n\ge n\sb0.$ Moreover, an a priori upper bound is given for $n\sb1$. This allows us to prove many hypergeometric identities by simply checking a finite (albeit large) number of initial values. With similar methods, we show explicit a priori upper bounds for $n\sb1$ in the cases where $\sum\sb{k}F(n,k) = f(n)$ (for some hypergeometric term $f(n))$ and $\sum\sb{k}F(n,k) = \sum\sb{k}G(n,k)$ (for some admissible proper-hypergeometric term $G(n,k))$ are the objects of interest. Finally, we generalize the above statement to the case of $\sum\sb{k\sb1}\sum\sb{k\sb2}\cdots\sum\sb{k\sb{r}} F(n,k\sb 1,k\sb2,\...,k\sb{r}) = 1.$

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Recommended Citation

Yen, Lily, "Contributions to the proof theory of hypergeometric identities" (1993). Dissertations available from ProQuest. AAI9331870.