By Jean-Paul Allouche and Doron Zeilberger
Published in The Ramanujan Journal v. 62, pages 189-214 (2023)
To the memory of Vladimir Shevelev (Mar 09, 1945 - May 03, 2018)
Written: April 15, 2022
Abstract: We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results that can be found in the literature. Also we prove two conjectures given in that paper. After mentioning many other works dealing with identities for various trigonometric sums, we end this paper by describing an automated approach for proving such trigonometric identities.
The input file generates the output file
The input file generates the output file
The input file generates the output file
COMMENT: THIS FILE IS NOT REALLY NEEDED, SINCE IT CAN BE SHOWN THAT For ALL i
Sum(sin(Pi*j/(2*n+1))^(2*i),j=1..n)=1/4^i*binomial(2*i,i)*(n+1/2)
The input file generates the output file
The input file generates the output file
The input file generates the output file
The input file generates the output file
COMMENT: THIS FILE IS NOT REALLY NEEDED, SINCE IT CAN BE SHOWN THAT For ALL i
Sum(sin(Pi*(2*j-1)/(4*n))^(2*i),j=1..n)=1/2^(2*i-1)*binomial(2*i-1,i-1)*n
The input file generates the output file
The input file generates the output file
The input file generates the output file
Note that it took about the same time.
The input file generates the output file
The input file generates the output file
The input file generates the output file
The input file generates the output file
The input file generates the output file