부산대학교 도서관

In 1890, academician A. A. Markov employed an ad-hoc method for constructing (in today's terminology) WZ-pairs which he needed to accelerate convergence of certain slowly-convergent numerical series (notably the one defining $\zeta(3)$). His method consists of starting with a proper hypergeometric kernel $H(x,z)$, then trying to determine functions $p(z)=P(x,z)$ and $q(z)=Q(x,z)$ (polynomials in $z$, with coefficients depending on $x$) in such a way that $U(x,z)=H(x,z)P(x,z)$ and $V(x,z)=H(x,z)Q(x,z)$ form a WZ-pair (meaning that $U(x+1,z)-U(x,z)=V(x,z+1)-V(x,z)$); this leads to a system of first-order linear recurrence equations with polynomial coefficients for the unknown coefficients of $p(z)$ and $q(z)$. \par The authors noticed that Markov's method can be combined with the parametric Gosper algorithm to yield an algorithm which will output the desired $p(z)=\sum_{i=0}^da_i(x) z^i$ provided that we allow $q(z)$ to be a rational function in $z$. Given $U(x+1,z)-U(x,z)$ as input, Gosper's algorithm (with respect to $z$) yields a system of linear equations relating $a_0(x)$, $\dots$, $a_d(x)$, $a_0(x+1)$, $\dots$, $a_d(x+1)$, and the coefficients of Gosper's polynomial. If the system contains sufficiently many linearly independent equations, the latter coefficients can be eliminated to yield a system of first-order linear recurrence equations with polynomial coefficients for the $a_i(x)$. It is plausible that by tracing the algorithm on a generic proper hypergeometric kernel, one can prove that for $d$ large enough the algorithm is guaranteed to succeed. The authors provide Maple packages MarkovWZ and its continuous counterpart ContMarkovWZ together with examples of convergence-acceleration formulas and new summation identities, freely available from the second author's webpage. They express hope that their algorithm may contribute to the effort of proving irrationality of constants such as $\zeta(5)$ and $\zeta(7)$.