TY - JOUR

T1 - On linear relations for dirichlet series formed by recursive sequences of second order

AU - Elsner, Carsten

AU - Technau, Niclas

N1 - Funding Information:
We thank the anonymous referee for valuable suggestions and careful reading. The second author was partially supported by the Austrian Science Fund (FWF) projects W1230 and Y-901 and EPSRC grant EP/J018260/1.
Publisher Copyright:
© 2020 Australian Mathematical Publishing Association Inc.

PY - 2021/6

Y1 - 2021/6

N2 - Let Fn and Ln and be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in are defined by As a consequence of Nesterenko's proof of the algebraic independence of the three Ramanujan functions R(ρ);Q(ρ); and P(ρ) for any algebraic number ρ with 0< ρ < 1, the algebraic independence or dependence of various sets of these numbers is already known for positive even integers. In this paper, we investigate linear forms in the above zeta functions and determine the dimension of linear spaces spanned by such linear forms. In particular, it is established that for any positive integer m the solutions of with ts; us; vs; ws ϵ Q (1 < s ≤ m) form a Q-vector space of dimension. This proves a conjecture from the Ph.D. thesis of Stein, who, in 2012, was inspired by the relation -2ζF(2) + ζ∗F(2) + 5ζ∗L(2) = 0. All the results are also true for zeta functions in, where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second-order recurrence formula.

AB - Let Fn and Ln and be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in are defined by As a consequence of Nesterenko's proof of the algebraic independence of the three Ramanujan functions R(ρ);Q(ρ); and P(ρ) for any algebraic number ρ with 0< ρ < 1, the algebraic independence or dependence of various sets of these numbers is already known for positive even integers. In this paper, we investigate linear forms in the above zeta functions and determine the dimension of linear spaces spanned by such linear forms. In particular, it is established that for any positive integer m the solutions of with ts; us; vs; ws ϵ Q (1 < s ≤ m) form a Q-vector space of dimension. This proves a conjecture from the Ph.D. thesis of Stein, who, in 2012, was inspired by the relation -2ζF(2) + ζ∗F(2) + 5ζ∗L(2) = 0. All the results are also true for zeta functions in, where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second-order recurrence formula.

KW - elliptic functions

KW - Fibonacci and Lucas numbers

KW - linear independence

KW - q-series

UR - http://www.scopus.com/inward/record.url?scp=85078305201&partnerID=8YFLogxK

U2 - 10.1017/S1446788719000491

DO - 10.1017/S1446788719000491

M3 - Article

AN - SCOPUS:85078305201

SN - 1446-7887

VL - 110

SP - 406

EP - 430

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

IS - 3

ER -