Apéry-like sequences defined by four-term recurrence relations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914787949019136 |
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| author | Cooper, Shaun |
| author_facet | Cooper, Shaun |
| contents | The Apéry numbers may be defined by a cubic three-term recurrence relation, that is, a three-term relation where the coefficients are polynomials in the index of degree $3$.
In this work, we first provide a systematic review of Apéry numbers and other related sequences that satisfy quadratic or cubic three-term recurrence relations, and show how they are interrelated and how they may be classified. This leads to sequences defined by cubic $k$-term recurrence relations. The cases corresponding to $k=2$ in this framework lead to Ramanujan's theories of elliptic functions to alternative bases, while the cases corresponding to $k=3$ correspond to the Apéry, Domb, Almkvist--Zudilin numbers and other sequences that are well-studied.
We conduct a detailed analysis for the case $k=4$. Some of the sequences that arise are new. Of particular interest are ten sequences that are said to be self-starting in the sense that a single initial condition is enough to start the recurrence relation. Of additional interest are two sequences which take values in $\mathbb{Z}[i]$ and two others with values in $\mathbb{Z}[\sqrt{2}]$. Congruence properties and asymptotic expansions for the ten self-starting sequences are investigated and several conjectures are presented. For example, we conjecture that the integer-valued sequence defined by the recurrence relation \begin{align*} (n+1)^3T(n+1) &=2(2n+1)(5n^2+5n+2)T(n) \\ &\qquad -8n(7n^2+1)T(n-1)+22n(2n-1)(n-1)T(n-2) \end{align*} and initial condition $T(0)=1$ satisfies a Lucas congruence for every prime $p$. Moreover, the sequence is conjectured to satisfy the supercongruence $$ T(pn) \equiv T(n) \pmod{p^2} \quad\text{for all positive integers $n$} $$ if $p=2,\;59$ or $5581$, and for no other primes $p<10^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00757 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Apéry-like sequences defined by four-term recurrence relations Cooper, Shaun Number Theory Primary 11F11, 33-02, Secondary 05A10, 11A07, 11B65, 11F20, 33C20 The Apéry numbers may be defined by a cubic three-term recurrence relation, that is, a three-term relation where the coefficients are polynomials in the index of degree $3$. In this work, we first provide a systematic review of Apéry numbers and other related sequences that satisfy quadratic or cubic three-term recurrence relations, and show how they are interrelated and how they may be classified. This leads to sequences defined by cubic $k$-term recurrence relations. The cases corresponding to $k=2$ in this framework lead to Ramanujan's theories of elliptic functions to alternative bases, while the cases corresponding to $k=3$ correspond to the Apéry, Domb, Almkvist--Zudilin numbers and other sequences that are well-studied. We conduct a detailed analysis for the case $k=4$. Some of the sequences that arise are new. Of particular interest are ten sequences that are said to be self-starting in the sense that a single initial condition is enough to start the recurrence relation. Of additional interest are two sequences which take values in $\mathbb{Z}[i]$ and two others with values in $\mathbb{Z}[\sqrt{2}]$. Congruence properties and asymptotic expansions for the ten self-starting sequences are investigated and several conjectures are presented. For example, we conjecture that the integer-valued sequence defined by the recurrence relation \begin{align*} (n+1)^3T(n+1) &=2(2n+1)(5n^2+5n+2)T(n) \\ &\qquad -8n(7n^2+1)T(n-1)+22n(2n-1)(n-1)T(n-2) \end{align*} and initial condition $T(0)=1$ satisfies a Lucas congruence for every prime $p$. Moreover, the sequence is conjectured to satisfy the supercongruence $$ T(pn) \equiv T(n) \pmod{p^2} \quad\text{for all positive integers $n$} $$ if $p=2,\;59$ or $5581$, and for no other primes $p<10^4$. |
| title | Apéry-like sequences defined by four-term recurrence relations |
| topic | Number Theory Primary 11F11, 33-02, Secondary 05A10, 11A07, 11B65, 11F20, 33C20 |
| url | https://arxiv.org/abs/2302.00757 |