부산대학교 도서관

In 1890, academician A. A. Markov employed an ad-hoc method forconstructing (in today's terminology) WZ-pairs which he needed toaccelerate convergence of certain slowly-convergent numerical series(notably the one defining $\zeta(3)$). His method consists of startingwith a proper hypergeometric kernel $H(x,z)$, then trying to determinefunctions $p(z)=P(x,z)$ and $q(z)=Q(x,z)$ (polynomials in $z$, withcoefficients 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 toa system of first-order linear recurrence equations with polynomialcoefficients for the unknown coefficients of $p(z)$ and $q(z)$.\parThe authors noticed that Markov's method can be combined with theparametric Gosper algorithm to yield an algorithm which will outputthe 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 $U(x+1,z)-U(x,z)$ asinput, Gosper's algorithm (with respect to $z$) yields a system of linearequations 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 systemcontains sufficiently many linearly independent equations, the lattercoefficients can be eliminated to yield a system of first-order linearrecurrence equations with polynomial coefficients for the $a_i(x)$. Itis plausible that by tracing the algorithm on a genericproper hypergeometric kernel, one can prove that for $d$ large enoughthe algorithm is guaranteed to succeed. The authors provideMaple packages MarkovWZ and its continuous counterpartContMarkovWZ together with examples of convergence-accelerationformulas and new summation identities, freely available from thesecond author's webpage. They express hope that their algorithm maycontribute to the effort of proving irrationality of constants such as$\zeta(5)$ and $\zeta(7)$.