In 1992, Wilf and Zeilberger conjectured that a hypergeometric term in several discrete and continuous variables is holonomic if and only if it is proper. Strictly speaking the conjecture does not hold, but it is true when reformulated properly: Payne proved a piecewise interpretation in 1997, and independently, Abramov and Petkovsek in 2002 proved a conjugate interpretation. Both results address the pure discrete case of the conjecture. In this paper we extend their work to hypergeometric terms in several discrete and continuous variables and prove the conjugate interpretation of the Wilf-Zeilberger conjecture in this mixed setting.
A Wilf--Zeilberger pair $(F, G)$ in the discrete case satisfies the equation $ F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k)$. We present a structural description of all possible rational Wilf--Zeilberger pairs and their continuous and mixed analogues.
In this paper, we prove some supercongruences via the Wilf-Zeilberger method. For instance, for any odd prime $p$ and positive integer $r$ and $δ\in\{1,2\}$, we have \begin{align*} \sum_{n=0}^{(p^r-1)/δ} \frac{\left(\frac12\right)^5_n}{n!^5}(10n^2+6n+1)(-4)^n &\equiv\begin{cases}p^{2r}\ \pmod{p^{r+4}} &\tt{if}\ r\leq4, \\0\ \pmod{p^{r+4}} &\tt{if}\ r \geq5. \end{cases} \end{align*}
In this paper, we prove two supercongruences by the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{3n+1}{(-8)^n}\binom{2n}n^3\equiv p\left(\frac{-1}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\left(\frac14\right)\pmod{p^4}, \end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol, and $E_{n}(x)$ are the Euler polynomials. This congruence confirms a conjecture of Sun \cite[(2.18)]{sun-numb-2019} with $n=1$.
By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $ζ(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such identity is then applied to show several supercongruences.
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{p-1}\frac{6n+1}{256^n}\binom{2n}n^3&\equiv p(-1)^{(p-1)/2}-p^3E_{p-3}\pmod{p^4}. \end{align*} In fact, this supercongruence is a generalization of a supercongruence of van Hamme.
Using Wilf-Zeilberger algorithmic proof theory, we continue pioneering work of Meni Rosenfeld (followed up by interesting work by Cyril Grunspan and Ricardo Perez-Marco) and study the probability and duration of successful bitcoin attacks, but using an equivalent, and much more congenial, formulation as a certain two-phase soccer match.
Linear homogeneous recurrence equations with polynomial coefficients are said to be holonomic. Such equations have been introduced in the last century for proving and discovering combinatorial and hypergeometric identities. Given a field K of characteristic zero, a term a(n) is called hypergeometric with respect to K, if the ratio a(n+1)/a(n) is a rational function over K. The solutions space of holonomic recurrence equations gained more interest in the 1990s from the well known Zeilberger's algorithm. In particular, algorithms computing the subspace of hypergeometric term solutions which cove…
Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$ \sum_{k=0}^{p-1} (3k-1)\frac{(\frac{1}{2})_k (-\frac{1}{2})_k^2 }{k!^3}4^k\equiv p-2p^3 \pmod{p^4}, $$ where $(a)_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer symbol.
