Let (Xn, Yn)n1 denote the positive integer solutions of Pell’s equation X2DY 2 = 1 or X2 DY 2 = 1. We introduce the Dirichlet series ?X(s) = P1 n=1 1/Xs n and ?Y (s) = P1 n=1 1/Y s n for 0 and prove that both functions do not satisfy any nontrivial algebraic di?erential equation. For any positive integers s1 and s2 the two numbers ?X(2s1) and ?Y (2s2) are algebraically independent over a transcendental field extension of Q, whereas the three numbers ?X(2), ?Y (2), and P1 n=1 1/(XnYn) 2 are linearly dependent over Q. From the transcendence of ?Y (2) and the corresponding alternating series we obtain an application to the Archimedean cattle problem. Irrationality results for series of the form P1 n=1 (1)n+1/Xn, P1 n=1 (1)n+1/Yn, and P1 n=1 (1)n+1/XnYn are obtained by a theorem of R. Andr′e-Jeannin.
展开▼
机译：让（xn，yn）n1表示Pell等式x2dy 2 = 1或x2 dy 2 = 1的正整数溶液。我们介绍了Dirichlet系列αx（s）= p1 n = 1 1 / xs n和Δy（s ）= p1 n = 1 1 / y sn为0并证明这两个功能不满足任何非活动代数di？潜列式等式。对于任何正整数S1和S2，两个数字？x（2s1）和Δy（2s2）在q的超越场延伸上是代数独立的，而三个数字Δx（2），Δy（2）和p1 n = 1 1 /（xnyn）2线性地依赖于Q.来自ΔY（2）的超越和相应的交替系列，我们获得了Archimedean牛问题的应用。通过a获得的P1 n = 1（1）n + 1 / xn，p1 n = 1（1）n + 1 / yn，p1 n = 1（1）n + 1 / xnyn的串联的非理性结果R. Andr'e-Jeannin的定理。
展开▼
