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On the linear independence of $p$-adic polygamma values

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Makoto Kawashima, Anthony Poëls

Number Theory · Submitted 9 Oct 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz zeta function $\zeta_p(s,x)$ at distinct shifts $x$.

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arXiv ID2410.06789 (version 1)
Pith IDGNXNBMEY, Pith's permanent record of this paper and its review
CitationAnthony Po\"els, Makoto Kawashima. "On the linear independence of $p$-adic polygamma values." arXiv:2410.06789 (2024). https://pith.science/paper/GNXNBMEY
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,283 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

In this article, we present a new linear independence criterion for values of the $p$-adic polygamma functions defined by J.~Diamond. As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz zeta function $\zeta_p(s,x)$ at distinct shifts $x$. This improves and extends a previous result due to P.~Bel [5], as well as irrationality results established by F.~Beukers [7]. Our proof is based on a novel and explicit construction of Pad\'{e}-type approximants of the second kind of Diamond's $p$-adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this article, we present a new linear independence criterion for values of the $p$-adic polygamma functions defined by J.~Diamond. contribution 0.99 0.93 0.18 asserted 1.00
s1 As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz zeta function $\zeta_p(s,x)$ at distinct shifts $x$. result 0.98 0.95 0.38 asserted 1.00
s2 This improves and extends a previous result due to P.~Bel [5], as well as irrationality results established by F.~Beukers [7]. interpretation 0.47 0.62 0.16 asserted 1.00
s3 Our proof is based on a novel and explicit construction of Pad\'{e}-type approximants of the second kind of Diamond's $p$-adic polygamma functions. method 0.94 0.95 0.04 asserted 1.00
s4 This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials. method 0.98 0.88 0.03 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this article, we present a new linear independence criterion for values of the $p$-adic polygamma functions defined by J.~Diamond. As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz zeta function $\zeta_p(s,x)$ at distinct shifts $x$. This improves and extends a previous result due to P.~Bel [5], as well as irrationality results established by F.~Beukers [7]. Our proof is based on a novel and explicit construction of Pad\'{e}-type approximants of the second kind of Diamond's $p$-adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 4 sentences without a confident label

Uses the method or tool of 1 paper
Cites as background 1 paper
  • Yu [ref] , and Fischler [ref] for irrationality and linear independence results of odd values of the Riemann zeta function.

    2109.10136 · Introduction · 0.94

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