Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Generating function identities for ζ(2n + 2), ζ(2n + 3) via the WZ method. | Generating function identities for 2n 2 2n 3 via the WZ method Kh. Hessami Pilehrood and T. Hessami Pilehroody Mathematics Department Faculty of Science Shahrekord University Shahrekord . Box 115 Iran Institute for Studies in Theoretical Physics and Mathematics IPM Tehran Iran hessamik@ hessamit@ Submitted Nov 25 2007 Accepted Feb 19 2008 Published Feb 29 2008 Mathematics Subject Classifications 11M06 05A10 05A15 05A19 Abstract Using WZ-pairs we present simpler proofs of Koecher Leshchiner and Bailey-Borwein-Bradley s identities for generating functions of the sequences 2n 2 n 0 and 2n 3 n 0. By the same method we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function. 1 Introduction The Riemann zeta function is defined by the series c s X i for Re s 1 n n 1 Apery s irrationality proof of c 3 and series acceleration formulae for the first values of the Riemann zeta function going back to Markov s work 8 1 c 2 3E i k 1 K k 5 _1 k 1 c 1 3 IX ựíỹ k 1 k k c 4 36 X_1_ c 17 u k4 2k k 1 k This research was in part supported by a grant from IPM No. 86110025 yThis research was in part supported by a grant from IPM No. 86110020 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R35 1 stimulated intensive search of similar formulas for other values n n 5. Many Aperylike formulae have been proved with the help of generating function identities see 6 1 5 11 4 . M. Koecher 6 and independently Leshchiner 7 proved that y 2k 3 a2k y 1 y T 1 1 5 k2 a2 n 1 - n n2 a2 2 k3 2 k k2 a2 11 m2 k 0 n 1 7 k 1 k m 1 n 1 for any a 2 C with a 1. For even zeta values Leshchiner 7 in an expanded form showed that see 4 31 1 z X 1 - k 0 1 1 y 2 2k r 2k 2 a2k 1 y n 1 -1 n-1 n2 a2 k 1 1 k2 2k k 3k2 a2 k2 - a2 n 1 - X m m 1 x 7 2 for any complex a with a 1. Recently D. Bailey J. Borwein and D. Bradley 4 proved another formula y 2k 2 0 y 1 3 y .1 - TT m 4a2 Ì zL 2k 2 a zL n2 _ a2 6 2k k2 _ 2 H m2 _ a2 k 0 n 1
