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Computing with Hypergeometric-Type Terms

Bertrand Teguia Tabuguia

TL;DR

Two algorithms are presented for computing holonomic recurrence equations from hypergeometric-type normal forms and for finding products of hypergeometric-type terms using newly implemented commands in the Maple package HyperTypeSeq.

Abstract

Take a multiplicative monoid of sequences in which the multiplication is given by Hadamard product. The set of linear combinations of interleaving monoid elements then yields a ring. For hypergeometric sequences, the resulting ring is a subring of the ring of holonomic sequences. We present two algorithms in this setting: one for computing holonomic recurrence equations from hypergeometric-type normal forms and the other for finding products of hypergeometric-type terms. These are newly implemented commands in our Maple package $HyperTypeSeq$, available at \url{https://github.com/T3gu1a/HyperTypeSeq}, which we also describe.

Computing with Hypergeometric-Type Terms

TL;DR

Two algorithms are presented for computing holonomic recurrence equations from hypergeometric-type normal forms and for finding products of hypergeometric-type terms using newly implemented commands in the Maple package HyperTypeSeq.

Abstract

Take a multiplicative monoid of sequences in which the multiplication is given by Hadamard product. The set of linear combinations of interleaving monoid elements then yields a ring. For hypergeometric sequences, the resulting ring is a subring of the ring of holonomic sequences. We present two algorithms in this setting: one for computing holonomic recurrence equations from hypergeometric-type normal forms and the other for finding products of hypergeometric-type terms. These are newly implemented commands in our Maple package , available at \url{https://github.com/T3gu1a/HyperTypeSeq}, which we also describe.
Paper Structure (4 sections, 1 theorem, 12 equations, 1 algorithm)

This paper contains 4 sections, 1 theorem, 12 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Recurrence Equations
  3. Products
  4. Conclusion

Key Result

Proposition 2

Let $j_1,j_2,m_1,m_2\in\mathbb{N}$ such that $j_1<m_1, j_2<m_2$. Let $\mu=\mathop{\mathrm{lcm}}\nolimits(m_1,m_2)$ and Then $\mathcal{N}=\{j_0\}$ or $\mathcal{N}=\emptyset$ and we have

Theorems & Definitions (7)

  • Definition 1: Hypergeometric-type sequence
  • Example 1.1: OEIS https://oeis.org/A212579 (see teguia2023hypergeometric)
  • Example 1.2
  • Remark 2.1
  • Example 2.2
  • Proposition 2: see Lemma 2 in teguia2023hypergeometric
  • Example 3.1