Polynomial reduction for holonomic sequences and applications in π-series and congruences

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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.