Human and Automated approaches for finite trigonometric sums

By Jean-Paul Allouche and Doron Zeilberger

.pdf    Latex source

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.

# Maple package

• TrigSums.txt, a Maple package for handling certain trigonometric sums

# Sample Input and Output for TrigSums.txt

• If you want to see a paper with the sum of the i-th powers of {1/sin(Pi*j/(2*n+1))^2,j=1..n) for i from 1 to 50

The input file generates the output file

• If you want to see, more tersely the above list of polynomials in n, for i from 1 to 60

The input file generates the output file

• If you want to see a paper with the sum of the i-th powers of {sin(Pi*j/(2*n+1))^2,j=1..n) for i from 1 to 50

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)

• If you want to see a fully rigorous computer-proof of Proposition 10 of the paper

The input file generates the output file

• If you want to see a paper with the sum of the i-th powers of {1/sin(Pi*(2*j-1)/(4*n))^2,j=1..n) for i from 1 to 50

The input file generates the output file

• If you want to see, more tersely the above list of polynomials in n, for i from 1 to 60

The input file generates the output file

• If you want to see a paper with the sum of the i-th powers of {sin(Pi*(2*j-1)/(4*n))^2,j=1..n}, for i from 1 to 50

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

• If you want to see a paper with the sum of the i-th powers of {csc(Pi*j/(2*n))^2, j=1..n-1},

The input file generates the output file

• If you want to see a file with the polynomials in n, that are the the sum of the i-th powers of {csc(Pi*j/(2*n)^2, j=1..n-1}, for i from 1 to 60 (same information as the above file, but going all the way to the sum of the 60-th powers)

The input file generates the output file

• If you want to see the same as the above, but using instead the Chu-Marini generating function

The input file generates the output file

Note that it took about the same time.

• If you want to see a paper with the sum of the i-th powers of {sin(Pi*j/(2*n))^2, j=1..n-1},

The input file generates the output file

• If you want to see a paper with the sum of the i-th powers of {csc(Pi*(2*j-1)/(4*n+2))^2,j=1..n}

The input file generates the output file

• If you want to see a file with the polynomials in n, that are the the sum of the i-th powers of {csc(Pi*(2*j-1)/(4*n+2))^2, j=1..n}, for i from 1 to 60 (same information as the above file, but going all the way to the sum of the 60-th powers)

The input file generates the output file

• If you want to see a paper with the sum of the i-th powers of {sin(Pi*(2*j-1)/(4*n+2))^2,j=1..n}

The input file generates the output file

• If you want to see Maple proofs of the 9 identities in the interesting paper Non-trivial trigonmetric sums arising from some of Ramanujan Theta function identities by G. Vinay, H. T. Shwetha and K. N. Harshitha

The input file generates the output file