부산대학교 도서관

A pair $(H(x,z), G(x,z))$ of bivariate hypergeometric terms is called a Markov-WZ pair if there is a function $P(x,z)$, polynomial in $z$, such that $$H(x+1,z)P(x+1,z)-H(x,z)P(x,z)=G(x,z+1)-G(x,z).$$ It is shown (with a few typos in the proof) that for any $H(x,z)$ that can be written as a ratio of products of factorials, there exists a $G(x,z)$ such that $(H(x,z),G(x,z))$ is a Markov-WZ pair, and an upper bound for the degree in $z$ of the corresponding $P(x,z)$ is given. \par This can be used to convert a given hypergeometric series into a different one. As an application, several rapidly converging series representations of the constants $\log 2$, $\zeta(2)$, and $\zeta(3)$ are given.