Applications of Caputo operators in the evaluation of Clebsch–Gordan-type multiple elliptic integrals

Abstract

We apply a method of semi-integration by parts (SIBP) that we had previously derived and formulated using the semi-derivative and semi-primitive operators. We obtain many new results on integrals involving the complete elliptic integrals as integrand factors, building on the work of Glasser, Cantarini, Wan, and Zhou. Our main results have not appeared in past literature concerning integrals involving products of the complete elliptic integral(s) of the first and/or second kinds. Furthermore, many previously known identities for such integrals are special cases of our SIBP identity for analytic functions.

Keywords:

• Integral transform
• fractional operator
• complete elliptic integral
• Clebsch–Gordan integral

Mathematics Subject Classifications:

• 33C75
• 26A33
• 33E05

Acknowledgements

The author thanks Marco Cantarini and Jacopo D'Aurizio for many discussions. Jacopo had suggested the use of the semi-integration by parts method from [2] in order to reduce Chu's series in (5.1(5.1) $\sum _{n=1}^{\mathrm{\infty }}{(\genfrac{}{}{0ex}{}{2n}{n})}^2\frac{H_n^{(2)}}{{32}^n}.$(5.1) ) to an elementary integral.

Disclosure statement

No potential conflict of interest was reported by the author.