LI’S RELATION INVOLVING THE NUMBER OF REPRESENTATIONS OF AN INTEGER AS A SUM OF SQUARES AND APPLICATION IN DIFFERENCE EQUATIONS
By
J. D. Bulnes1, R. C. Singh Chandel2 , Hemant Kumar3 , P. Siva Kota Reddy4 and J. L´opez-Bonilla5
1Departamento de Ciencias Exatas e Tecnolog´ıa, Universidade Federal do Amap´a, Rod. Juscelino Kubitschek, Jardin Marco Zero, Macap´a, AP, Brasil-68903-419.
2Former Head, Department of Mathematics, D. V. Postgraduate College Orai, Uttar Pradesh, India-285001.
3Department of Mathematics, D. A-V. Postgraduate College, Kanpur, Uttar Pradesh, India-208001.
4Department of Mathematics, JSS Science and Technology University, Mysuru, India-570 006.
5ESIME-Zacatenco, Instituto Polit´ecnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista, CDMX, M´exico, CP-07738.
Email: bulnes@unifap.br, rc chandel@yahoo.com, palhemant2007@rediffmail.comm hemantkumar kn03@csjmu.ac.in, pskreddy@jssstuniv.in, pskreddy@sjce.ac.in, jlopezb@ipn.mx
(Received: February 06, 2025; In format: March 16, 2025; Revised: September 18, 2025; Accepted: September 22, 2025)
DOI: https://doi.org/10.58250/jnanabha.2025.55205
Abstract
The formulae for the sum of inverses of odd divisors of an integer n are deduced by various authors in the literature. Here, in this paper, we exhibit that their expressions are related by certain identity of Li involving the number of representations of n as a sum of squares.
2020 Mathematical Sciences Classification: 05A17, 05A20, 33C45, 11E25.
Keywords and Phrases: Stirling numbers, Li’s expression, Integers as a sum of squares, Jha’s and Glaisher’s relations, Sum of inverses of odd divisors, Binomial transform, Bell polynomials, Theta functions.