Explicit evaluation of some families of log-sine integrals via the hypergeometric mechanism and their applications

Shakir Hussain Malik and Mohammad Idris Qureshi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 52–75
DOI: 10.7546/nntdm.2026.32.1.52-75
Full paper (PDF, 312 Kb)

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Authors and affiliations

Shakir Hussain Malik
Department of Humanities and Basic Sciences, Sri Indu College of Engineering and Technology (An Autonomous Institution under UGC, New Delhi)
Sheriguda (V), Ibrahimpatnam, Ranga Reddy District, Hyderabad, Telangana, 501510, India

Mohammad Idris Qureshi
Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University)
New Delhi, 110025, India

Abstract

In this paper, we present explicit analytical expressions for certain families of log-sine definite integrals: , expressed in terms of multiple hypergeometric functions of the Kampé de Fériet with arguments and the Riemann zeta functions. As applications, we establish several mixed summation formulas (79), (81) and (83) involving the generalized hypergeometric functions , and . Moreover, a collection of possibly new summation formulas (42), (52), (54), (56), (58), (62), (64), (66), (70), (72), (74) and (76) for multiple hypergeometric functions of the Kampé de Fériet are derived. In addition, mixed relations (80), (82) and (84) involving the Riemann zeta functions are also established.

Keywords

• Log-sine integrals
• Generalized hypergeometric functions
• Riemann zeta function
• Kampé de Fériet functions

2020 Mathematics Subject Classification

• 33C05
• 33C20
• 11M06
• 26A42

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Manuscript history

• Received: 29 October 2025
• Revised: 23 November 2025
• Accepted: 10 December 2025
• Online First: 19 February 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).