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Recently, Hou, Mu and Zeilberger introduced a new process of polynomial reduction for hypergeometric terms, which can be used to prove and generate hypergeometric identities automatically. In this paper, we extend this polynomial reduction to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of π -series involving Domb numbers and Franel numbers, and new families of congruences for Franel numbers and Delannoy numbers. [ABSTRACT FROM AUTHOR]

Let Δ x f ( x , y ) = f ( x + 1 , y ) − f ( x , y ) and Δ y f ( x , y ) = f ( x , y + 1 ) − f ( x , y ) be the difference operators with respect to x and y . A rational function f ( x , y ) is called summable if there exist rational functions g ( x , y ) and h ( x , y ) such that f ( x , y ) = Δ x g ( x , y ) + Δ y h ( x , y ) . Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method in the sense of algorithms, we need to solve two problems. The first is to determine the shift equivalence of two bivariate polynomials. We solve this problem by presenting an algorithm for computing the dispersion sets of any two bivariate polynomials. The second is to solve a univariate difference equation in an algebraically closed field. By considering the irreducible factorization of the denominator of f ( x , y ) in a general field, we present a new criterion which requires only finding a rational solution of a bivariate difference equation. We give a new estimation of the universal denominators based on the m -fold Gosper representation and transform the bivariate difference equation to a system of linear difference equations in one variable. Combining these algorithms, we can decide the summability of a bivariate rational function. [ABSTRACT FROM AUTHOR]

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach. [ABSTRACT FROM AUTHOR]