ILLINOIS JOURNAL OF MATHEMATICS Volume 37, Number 2, Summer 1993 ON SOME STRANGE SUMMATION FORMULAS R. WILLIAM GOSPER, MOURAD E.H. ISMAIL AND RUIMING ZHANG 1. Introduction During the last two years the first named author used symbolic algebra programs and long hours of computer experiments to formulate several infinite series identities. Some of his conjectures were communicated to other mathematicians as informal letters, and were circulated among interested parties. In particular [10] and [11] contained several conjectures in the form of identities reminiscent of Ramanujan’s work. Some of the series relevant to this work are ( 1.1) E ( 1)" : 2n n=l n2 cos + "rr ) - b 3 7r 2 (1.2) E (- 1)" cos(V/n27r2 9) 12e3 n=l //2 (1.3) n (- 1) siny/b2 + r2(n + 1/2)2 7r sin b - 1/2)2 2 b n+ +,wE(n+ (1.4) E cs[v/(nTr + b + a/b)(n,rr + dp + ab)] ( 1)n( )2 n’rr + b ( b + 1/b)sin a COS a cot b sin b 2 sin b Received June 14, 1991. 1991 Mathematics Subject Classification. Primary 33A10, 30B10; Secondary 33A40, 30B50. Research partially supported by the National Science Foundation. (C) 1993 by the Board of Trustees of the University of Illinois Manufactured in the United States of America 240