Hermite’s approach to Abelian integrals revisited
Makoto Kawashima1
1Totsuka, Yokohama, Kanagawa, 224-8539, Japan
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 557-588

In this article, we establish a new linear independence criterion for the values of certain Lauricella hypergeometric series $F_D$ with rational parameters, in both the complex and $p$-adic settings, over an algebraic number field. This result generalizes a theorem of C. Hermite [14] on the linear independence of certain Abelian integrals. Our proof relies on explicit Padé-type approximations to solutions of a reducible Jordan–Pochhammer differential equation, which extends the Padé approximations for certain Abelian integrals in [14]. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Padé-type approximants.

Dans cet article, nous établissons un nouveau critère d’indépendance linéaire pour les valeurs de certaines séries hypergéométriques de Lauricella $F_D$ à paramètres rationnels, aussi bien dans le cadre complexe que $p$-adique, sur un corps de nombres algébriques. Ce résultat généralise un théorème de C. Hermite [14] sur l’indépendance linéaire de certaines intégrales abéliennes. Notre démonstration repose sur des approximations de type Padé explicites pour les solutions d’une équation différentielle de Jordan–Pochhammer réductible, ce qui étend les approximations de Padé pour certaines intégrales abéliennes de [14]. La principale nouveauté de notre approche réside dans la preuve de la non-annulation des déterminants associés à ces approximants de type Padé.

Reçu le :
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DOI : 10.5802/jtnb.1373
Classification : 11J72, 33C65, 41A21
Keywords: Padé approximants, Rodrigues formula, linear independence, Lauricella hypergeometric series, Jordan–Pochhammer differential equation, $G$-functions, linear independence

Makoto Kawashima  1

1 Totsuka, Yokohama, Kanagawa, 224-8539, Japan
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Makoto Kawashima. Hermite’s approach to Abelian integrals revisited. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 557-588. doi: 10.5802/jtnb.1373
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[1] Yves André ${G}$-functions and geometry, Aspects of Mathematics, 13, Vieweg & Sohn, 1989, xii+229 pages | Zbl | DOI | MR

[2] Yves André Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité, Ann. Math. (2), Volume 151 (2000) no. 2, pp. 705-740 | Zbl | MR

[3] Yves André Arithmetic Gevrey series and transcendence. A survey, J. Théor. Nombres Bordeaux, Volume 15 (2003) no. 1, pp. 1-10 | Zbl | Numdam | DOI | MR

[4] Alexander I. Aptekarev; Amilcar Branquinho; Walter Van Assche Multiple orthogonal polynomials for classical weights, Trans. Am. Math. Soc., Volume 355 (2003) no. 10, pp. 3887-3914 | Zbl | DOI

[5] Paul Bateman; Jeffrey Kalb; Allen Stenger Problem 10797: A limit involving least common multiples, Am. Math. Mon., Volume 109 (2002) no. 4, pp. 393-394 | Zbl | DOI | MR

[6] Frits Beukers Irrationality of some $p$-adic ${L}$-values, Acta Math. Sin., Engl. Ser., Volume 24 (2008) no. 4, pp. 663-686 | Zbl | MR | DOI

[7] David V. Chudnovsky; Gregory V. Chudnovsky Use of computer algebra for Diophantine and differential equations, Computer Algebra (New York, 1984) (Lecture Notes in Pure and Applied Mathematics), Volume 113, Marcel Dekker, 1984, pp. 1-81 | Zbl

[8] Sinnou David; Noriko Hirata-Kohno; Makoto Kawashima Can polylogarithms at algebraic points be linearly independent?, Mosc. J. Comb. Number Theory, Volume 9 (2020) no. 4, pp. 389-406 | Zbl | DOI | MR

[9] Sinnou David; Noriko Hirata-Kohno; Makoto Kawashima Linear Forms in Polylogarithms, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 23 (2022) no. 3, pp. 1447-1490 | Zbl | MR

[10] Sinnou David; Noriko Hirata-Kohno; Makoto Kawashima Linear independence criteria for generalized polylogarithms with distinct shifts, Acta Arith., Volume 206 (2022) no. 2, pp. 127-169 | Zbl | DOI | MR

[11] Sinnou David; Noriko Hirata-Kohno; Makoto Kawashima Linear independence of values of hypergeometric functions and arithmetic Gevrey series (2025) (to appear in Acta Arith.) | arXiv

[12] Stéphane Fischler; Tanguy Rivoal A note on ${G}$-operators of order 2, Colloq. Math., Volume 170 (2022) no. 2, pp. 321-340 | Zbl | DOI | MR

[13] Charles Hermite Sur la fonction exponentielle, C. R. Acad. Sci. Paris, Volume 77 (1873), pp. 18-24 | Zbl

[14] Charles Hermite Lettre de M. C. Hermite de Paris à M. L. Fuchs de Göttingue “Sur quelques équations différentielles linéaires”, J. Reine Angew. Math., Volume 79 (1875), pp. 324-338 | Zbl | MR

[15] Marc Huttner On a paper of Hermite and Diophantine Approximation of Abelian Integrals, RIMS Kokyuroku, Volume 1219 (2001), pp. 185-194 (special issue “Analytic Number Theory: Expectations for the 21st Century”) | Zbl

[16] Marc Huttner On linear independence measure of some abelian integrals, Kyushu J. Math., Volume 57 (2003), pp. 129-157 | DOI | Zbl | MR

[17] Edward L. Ince Ordinary differential equations, Dover Publications, 1956

[18] Makoto Kawashima Rodrigues formula and linear independence for values of hypergeometric functions with parameters vary, J. Aust. Math. Soc., Volume 117 (2024) no. 3, pp. 308-344 | Zbl | DOI | MR

[19] Makoto Kawashima; Anthony Poëls Padé approximation for a class of hypergeometric functions and parametric geometry of numbers, J. Number Theory, Volume 243 (2023), pp. 646-687 | Zbl | DOI | MR

[20] Makoto Kawashima; Anthony Poëls On the linear independence of $p$-adic polygamma values, Mathematika, Volume 71 (2025) no. 4, e70040, 56 pages | DOI | Zbl | MR

[21] E. M. Nikišin; V. N. Sorokin Rational Approximations and Orthogonality, Translations of Mathematical Monographs, 92, American Mathematical Society, 1991, viii+221 pages | Zbl | DOI

[22] Henri Padé Sur la représentation approchée d’une fonction par des fractions rationnelles, Ann. Sci. Éc. Norm. Supér. (3), Volume 9 (1892), pp. 3-93 | Zbl | Numdam | DOI | MR

[23] Henri Padé Mémoire sur les développements en fractions continues de la fonction exponentielle, pouvant servir d’introduction à la théorie des fractions continues algébriques, Ann. Sci. Éc. Norm. Supér. (3), Volume 16 (1899), pp. 395-426 | Zbl | Numdam | DOI | MR

[24] Oskar Perron Über Summengleichungen und Poincarésche Differenzengleichungen, Math. Ann., Volume 84 (1921), pp. 1-15 | Zbl | DOI | MR

[25] Mihály Pituk More on Poincaré’s and Perron’s theorems for difference equations, Differ. Equ. Appl., Volume 8 (2002) no. 3, pp. 201-216 | Zbl | DOI | MR

[26] Georges Rhin; Philippe Toffin Approximants de Padé simultanés de logarithmes, J. Number Theory, Volume 24 (1986), pp. 284-297 | Zbl | DOI | MR

[27] Carl L. Siegel Über einige Anwendungen diophantischer Approximationen, Abh. Preuß. Akad. Wiss., Phys.-Math. Kl., Volume 1929 (1929) no. 1, pp. 1-70 (English transl. in On some applications of Diophantine approximations, Quaderni. Scuola Normale Superiore di Pisa. Monographs, vol. 2, Edizioni della Normale, 2014, p. 81-138) | Zbl

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