This book presents a comprehensive and focused attempt to derive key properties of multidimensional, or multi-index, Chebyshev polynomials by using generalized Hermite polynomials as a foundational tool. It demonstrates how multi-index Hermite polynomials can be employed to construct multidimensional Chebyshev polynomials of both the first and second kinds. Through symbolic and integral techniques, including a formal treatment of the Laplace transform, the book investigates various generalizations of these polynomial families. Emphasizing multi-index formulations, it explores these polynomials through a symbolic framework involving suitable integral transforms by leveraging the Laplace transforms. Key operational techniques are developed and applied to deepen conceptual understanding and navigate the formal structures underpinning various derived relationships.

The discussion is highlighted in applications inspired by real-world physical problems. Multi-index Hermite polynomials are examined in the context of quantum optics to model both coherent and incoherent radiation field distributions. Multidimensional systems coupled through electromagnetic radiation are addressed, alongside related wave propagation phenomena. Higher-order Laguerre polynomials are utilized to compute statistical moments of chaotic radiation, while multidimensional Bessel functions are explored for their role in laser theory. Traditional applications of Chebyshev polynomials in approximation theory are also revisited, providing a bridge between classic and contemporary mathematical approaches.

The content of the book is designed into three parts, each addressing a distinct facet of the subject. Part I is devoted to the algebraic theory of general set-theoretic solutions to the Yang–Baxter equation, with particular emphasis on skew left braces and Rota–Baxter groups. Part II presents a detailed treatment of the algebraic theory of racks and quandles. Part III, the most advanced part of the book, is concerned with the homology and cohomology theories associated with solutions to the Yang–Baxter equation. From the point of view of logical dependency, Parts I and II are largely self-contained and may be read independently, while Part III builds upon foundational concepts introduced in the earlier parts.

Clemente Cesarano is Associate Professor of Numerical Analysis at UNINETTUNO University, Italy, where he coordinates the local mathematics section and the doctoral course in technological innovation engineering. He has carried out teaching and research activities in various Italian and European institutions and universities. He has been Visiting Professor in some European universities, including the University of Linz, Austria, and the Complutense University of Madrid, Spain. He is the author of over 200 scientific publications and manuals in the field of approximation theory and mathematical analysis. He has participated, also as coordinator, in various national and international funded research projects.

Praveen Agarwal is Professor of Mathematics and Vice-Principal at Anand International College of Engineering, Jaipur, India. He holds a master’s degree from Rajasthan University (2000) and earned his Ph.D. from MNIT Jaipur (2006). Recognized among the world’s top 2% scientists for three consecutive years (2020–2022) by Stanford University, he has authored/edited over 12 books and published 350+ research papers with collaborators worldwide. In Research.com’s 2023 rankings, he placed 21st in India and 2436th globally in mathematics. With 23 years of experience, his research spans special functions, fractional calculus, numerical analysis, differential equations, and fixed-point theorems. He has served in esteemed roles at universities across Germany, Turkey, South Korea, the UK, Russia, Malaysia, and Thailand. His honors include ICMS Fellowship (UK), TUBITAK Scientist Award (Turkey), and Most Outstanding Researcher 2018 from India’s HRD Ministry. His work has over 7,500 citations (Google Scholar, h-index: 48) and 4,700+ citations (Scopus, h-index: 37).

Luis Vázquez is Full Professor of Applied Mathematics in the Department of Computer Sciences at the Complutense University of Madrid (UCM), where he earned his Licenciado in Physics in 1971. He completed his Ph.D. in Physics at Zaragoza University in 1975. His academic journey includes appointments as Visiting Research Associate at Brown University (1975–1977) and Scientific Associate at the Center for Nonlinear Studies, Los Alamos National Laboratory (1987–1990). At UCM, he served as Vice-Dean of the Faculty of Physics, Vice-President of Research, and Director of the European Office for Research. He founded and led the Supercomputation Center and coordinated scientific foresight at Spain’s National Agency for Evaluation and Foresight (ANEP). He is a founding member of the Centro de Astrobiología, affiliated with NASA’s Astrobiology Institute, and a Corresponding Academician of the Royal Academy of Exact, Physical and Natural Sciences of Spain. He has collaborated with institutions worldwide including CERN, NASA’s JPL, Bielefeld, Sinica Academy, Helsinki, Paris VI, Marseille, Bologna, Pisa, Goteborg, Kharkov, and several Spanish universities. He has served on expert committees for the EU’s R&D Framework Programs. He has authored 160 journal articles, 55 conference contributions, and 17 books, directed 12 Ph.D. theses, and participated in 45 research projects. His work spans nonlinear wave equations, numerical and stochastic methods, and complex systems. His current focus includes fractional calculus and Mars exploration, contributing to Beagle 2 (2003), REMS (2004–2007, 2011), ESA’s MiniHUM (2013), and MetNet (2011).