### Abstract

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 {x_{1}, . . . , x_{n}} of algebraic independent parameters by a system fi(x_{1}, . . . , x_{n}, 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, e^{π√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(q^{r}),Q(q^{r}), and R(q^{r}) 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.

Original language | English |
---|---|

Pages (from-to) | 121-141 |

Number of pages | 21 |

Journal | Functiones et Approximatio, Commentarii Mathematici |

Volume | 47 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 |

### Fingerprint

### Keywords

- Algebraic independence
- Complete elliptic integrals
- Gamma function
- Nesterenko's theorem
- Ramanujan functions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Functiones et Approximatio, Commentarii Mathematici*,

*47*(1), 121-141. https://doi.org/10.7169/facm/2012.47.1.10

**Algebraic independence of certain numbers related to modular functions.** / Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata.

Research output: Contribution to journal › Article

*Functiones et Approximatio, Commentarii Mathematici*, vol. 47, no. 1, pp. 121-141. https://doi.org/10.7169/facm/2012.47.1.10

}

TY - JOUR

T1 - Algebraic independence of certain numbers related to modular functions

AU - Elsner, Carsten

AU - Shimomura, Shun

AU - Shiokawa, Iekata

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

UR - http://www.scopus.com/inward/record.url?scp=84983430988&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84983430988&partnerID=8YFLogxK

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 -