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A note on functional equations of the \(p\)-adic polylogarithms. (English) Zbl 0748.12006

The author gives some sufficient and necessary conditions how to obtain a functional equation for polylogarithms. He discusses complex polylogarithms and \(p\)-adic polylogarithms [see R. F. Coleman, Invent. Math. 69, 171-208 (1982; Zbl 0516.12017)] as well. The main result is proved both by methods of differential geometry and differential algebra. The differential algebra proof is based on the fact that members of any functional equation in the polylogarithms may be discussed as the solutions of a system of linear differential equations with a nilpotent Galois group. So any algebraic dependence between the solutions of the system is a dependence in a ring of regular functions on a Galois group and it must have a special form [see M. F. Singer, Trans. Am. Math. Soc. 295, 753-763 (1986; Zbl 0593.12014)].

MSC:

12H05 Differential algebra
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
34A30 Linear ordinary differential equations and systems
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References:

[1] ABEL (N.H.) . - Note sur la fonction \?(x) = x + x\(^{2}\)/2\(^{2}\) + x\(^{3}\)/3\(^{2}\) + ... , Oeuvres, Bd. II, pp. 189-193.
[2] ANDRE (Y.) . - Quatre descriptions des groupes de Galois différentiels , Proceedings of the Séminaire d’algèbre de Paris, Lecture Notes in Math. 1296, Springer-Verlag. MR 89e:12012 | Zbl 0651.12015 · Zbl 0651.12015
[3] COLEMAN (R.F.) . - Dilogarithms , Regulators and p-adic L-functions, Invent. Math., t. 69, 1982 , p. 171-208. MR 84a:12021 | Zbl 0516.12017 · Zbl 0516.12017
[4] CARTIER (P.) . - Jacobiennes généralisées, monodromie unipotente et intégrales itérées , Séminaire Bourbaki, 1987 - 1988 , 687, Astérisque, pp. 161-162, 1988 . Numdam | Zbl 0688.14036 · Zbl 0688.14036
[5] CHEN (K.T.) . - Algebra of iterated path integrals and fundamental groups , Trans. Amer. Math. Soc., t. 156, 1971 , p. 359-379. MR 43 #1069 | Zbl 0217.47705 · Zbl 0217.47705
[6] DELIGNE (P.) . - Le groupe fondamental de la droite projective moins trois points , in Galois Groups over Q, Springer-Verlag New York, 1989 . MR 90m:14016 | Zbl 0742.14022 · Zbl 0742.14022
[7] DELIGNE (P.) . - Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs . · Zbl 0799.19004
[8] LEWIN (L.) . - Polylogarithms and Associated Functions . - North Holland, New York-Oxford, 1981 . MR 83b:33019 | Zbl 0465.33001 · Zbl 0465.33001
[9] LEWIN (L.) . - The order-independence of the polylogarithmic ladder structure, implications for a new category of functional equations , Aequationes Math., t. 30, 1986 , p. 1-20. MR 87i:33039 | Zbl 0606.39005 · Zbl 0606.39005
[10] MAGNUS (W.) , KARRASS (A.) and SOLITAR (D.) . - Combinatorial Group Theory , Pure and Applied Mathematics, volume XIII, Interscience Publishers, 1966 , John Wiley and Sons, Inc.. · Zbl 0132.17603
[11] ROGERS (L.J.) . - On function sum theorems connected with the series \infty \sum n=1 xn/n\(^{2}\) , Proc. London Math. Soc.(2), t. 4, 1907 , p. 169-189. JFM 37.0428.03 · JFM 37.0428.03
[12] SANDHAM (H.F.) . - A Logarithmic Transcendent , J. London Math. Soc., t. 24, 1949 , p. 83-91. MR 11,433b | Zbl 0036.32501 · Zbl 0036.32501
[13] WOJTKOWIAK (Z.) . - A note on the functional equation of the dilogarithm , preprint 1, CRM (Bellaterra), 1984 .
[14] WOJTKOWIAK (Z.) . - A note on functional equation of polylogarithms , preprint 22, MPI, 1990 .
[15] WOJTKOWIAK (Z.) . - The basic structure of polylogarithmic functional equations , preprint 88, MPI, 1990 .
[16] WOJTKOWIAK (Z.) . - A construction of analogs of the Bloch-Wigner function , Math. Scand., t. 65, 1989 , p. 140-142. MR 92b:11084 | Zbl 0698.33002 · Zbl 0698.33002
[17] ZAGIER (D.) . - Polylogarithms, Dedekind zeta function and the algebraic K-theory of fields , preprint 44, MPI, 1990 .
[18] ZAGIER (D.) . - The remarkable dilogarithm , J. Math. Phys. Sci., t. 22, 1988 , p. 131-145. MR 89f:33024 | Zbl 0669.33001 · Zbl 0669.33001
[19] ZAGIER (D.) . - The Bloch-Wigner-Ramakrishnan polylogarithm function , Math. Ann., t. 28, 1990 , p. 613-624. Article | MR 90k:11153 | Zbl 0698.33001 · Zbl 0698.33001
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