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• W1569661092 abstract "H. Wilf and D. Zeilberger (in Invent. Math. 108:575-633, 1992) developed a proof theory for hypergeometric multisums centered around the notion of a multivariate hypergeometric term. A multivariate function $f(zsb1,...,zsb{k})=f(z)$ from $doubzsp{k}$ to a field K is a hypergeometric term if for each $iin{ 1,...,k} $ there exist nonzero polynomials $Asb{i}(z)$ and $Bsb{i}(z)$ in $Klbrack zrbrack$ such that$$Asb{i}(z)f(z)=Bsb{i}(z)f(z+esb{i})$$for all $zindoubzsp{k}.$ Here $esb1,...,esb{k}$ is the standard basis for $doubzsp{k}.$ If f is not a zero-divisor, then the $term ratios Rsb1=Asb1/Bsb1,...,Rsb{k}=Asb{k}/Bsb{k}$ are unique and satisfy the relation$$Rsb{i}(z)Rsb{j}(z+esb{i})=Rsb{j}(z)Rsb{i}(z+esb{j})quad i,jin{ 1,...,k}.$$We introduce the concepts of divisibility lattice paths, rational Galois spaces, and fixed factors of rational functions to the study of the relation for the term ratios. We prove that a solution $Rsb1,...,Rsb{k}$ must be of the form$$Rsb{i}(z)={C(z+esb{i})over C(z)}{D(z)over D(z+esb{i})}prodlimitssb{vin V}prodsb{j}sbsp{0}{esb{i}cdot v}{asb{v}(vcdot z+j)over bsb{v}(vcdot z+j)} {rm for} iin{ 1,...,k},$$where C and D are polynomials, V is a finite subset of $doubzsp{k},$ and, for each $vin V, asb{v}$ and $bsb{v}$ are univariate polynomials all independent of i. The symbol $prodlimitssb{j}sbsp{a}{b}$ denotes $prodsbsp{i=a}{b-1}$ if $b>a$, 1 if $a=b,$ and 1/$prodsbsp{i=b}{a-1}$ if $a>b.$We use this factorization of $Rsb{i}$ to answer an obvious question about multivariate hypergeometric terms. Recall the Pochhammer symbol $(m)sb{r}=m(m+1)cdots(m+r-1).$ The multivariate hypergeometric terms that arise in practice have the form$$f(z)=gammasbsp{1}{zsb1}cdotsgammasbsp{k}{zsb{k}} {C(z)over D(z)}{prodsbsp{i=1}{p}(msb{i})sb{vsb{i}cdot z+rsb{i}}over prodsbsp{j=1}{q}(nsb{j})sb{wsb{j}cdot z+ssb{i}}}$$where $gammasb1,...,gammasb{k}in K, C$ and D are polynomials in $Klbrack zrbrack,$ the $vsb{i}$ and $wsb{j}$ are in $doubzsp{k},$ the $rsb{i}$ and $ssb{j}$ are in $doubz$, and the $msb{i}$ and $nsb{j}$ are in K. The question is: do all hypergeometric terms have this form? We prove that if K is algebraically closed, and the hypergeometric term is honest, then such an expression for f exists piecewise. This is trivial in the case of one variable, but not in the case of several variables. We use this result to settle the discrete part of a conjecture of Wilf and Zeilberger by showing that a holonomic hypergeometric term is piecewise proper, which means roughly that we can take $D(z)=1$ in the expression for f above." @default.
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• W1569661092 modified "2023-09-26" @default.
• W1569661092 title "Multivariate hypergeometric terms" @default.
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