Irrationality of certain numbers that contain values of the di - and trilogarithm

We prove that, for z ∈ {1/2, 2/3, 3/4, 4/5}, at least one of the two numbers ∞ ∑ l=1 zl l2 + log(1 − z) log z and ∞ ∑ l=1 zl l3 + 1 2 log(1 − z) log2 z is irrational. Mathematics Subject Classification (2000) 11J72 · 11G55 · 33C20 · 33C60 The irrationality result that we prove here, involves the values of the functions f j (z) = Li j (z)− Li1(z) · log j−1 z ( j − 1)! , j = 2, 3, . . . . Kh. Hessami Pilehrood was supported in part by Institute for Studies in Theoretical Physics and Mathematics (IPM) grant no. 83110021. T. Hessami Pilehrood was supported in part by IPM grant no. 83110020. W. Zudilin was supported in part by Russian Foundation for Basic Research grant no. 03-0100359. Kh. Hessami Pilehrood · T. Hessami Pilehrood School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran E-mail: [email protected], [email protected] Present address: Mathematics Department, Shahrekord University, P.O. Box 115, Shahrekord, Iran W. Zudilin (B) Department of Mechanics and Mathematics, Moscow Lomonosov State University, Vorobiovy Gory, GSP-2, 119992 Moscow, Russia E-mail: [email protected] 300 Kh. Hessami Pilehrood et al.