TY - JOUR

T1 - Algebraic independence of certain numbers related to modular functions

AU - Elsner, Carsten

AU - Shimomura, Shun

AU - Shiokawa, Iekata

N1 - Publisher Copyright:
© Wydawnictwo Naukowe UAM, Poznań 2012.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - In previous papers the authors established a method how to decide on the algebraic independence of a set {y1, . . . , yn} when these numbers are connected with a set {x1, . . . , xn} of algebraic independent parameters by a system fi(x1, . . . , xn, y1, . . . , yn) = 0 (i = 1, 2, . . . , n) of rational functions. Constructing algebraic independent parameters by Nesterenko's theorem, the authors successfully applied their method to reciprocal sums of Fibonacci numbers and determined all the algebraic relations between three q-series belonging to one of the sixteen families of q-series introduced by Ramanujan. In this paper we first give a short proof of Nesterenko's theorem on the algebraic independence of π, eπ√d and a product of Gamma-values Γ(m/n) at rational points m/n. Then we apply the method mentioned above to various sets of numbers. Our algebraic independence results include among others the coefficients of the series expansion of the Heuman-Lambda function, the values P(qr),Q(qr), and R(qr) of the Ramanujan functions P,Q, and R, for q ∈ Q with 0 < |q| < 1 and r = 1, 2, 3, 5, 7, 10, and the values given by reciprocal sums of polynomials.

AB - In previous papers the authors established a method how to decide on the algebraic independence of a set {y1, . . . , yn} when these numbers are connected with a set {x1, . . . , xn} of algebraic independent parameters by a system fi(x1, . . . , xn, y1, . . . , yn) = 0 (i = 1, 2, . . . , n) of rational functions. Constructing algebraic independent parameters by Nesterenko's theorem, the authors successfully applied their method to reciprocal sums of Fibonacci numbers and determined all the algebraic relations between three q-series belonging to one of the sixteen families of q-series introduced by Ramanujan. In this paper we first give a short proof of Nesterenko's theorem on the algebraic independence of π, eπ√d and a product of Gamma-values Γ(m/n) at rational points m/n. Then we apply the method mentioned above to various sets of numbers. Our algebraic independence results include among others the coefficients of the series expansion of the Heuman-Lambda function, the values P(qr),Q(qr), and R(qr) of the Ramanujan functions P,Q, and R, for q ∈ Q with 0 < |q| < 1 and r = 1, 2, 3, 5, 7, 10, and the values given by reciprocal sums of polynomials.

KW - Algebraic independence

KW - Complete elliptic integrals

KW - Gamma function

KW - Nesterenko's theorem

KW - Ramanujan functions

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U2 - 10.7169/facm/2012.47.1.10

DO - 10.7169/facm/2012.47.1.10

M3 - Article

AN - SCOPUS:84983430988

VL - 47

SP - 121

EP - 141

JO - Functiones et Approximatio, Commentarii Mathematici

JF - Functiones et Approximatio, Commentarii Mathematici

SN - 0208-6573

IS - 1

ER -