Classical and Quantum Orthogonal Polynomials in One Variable by Mourad E. H. Ismail – A Comprehensive Reference on Speci

Introduction

Orthogonal polynomials form a cornerstone of modern mathematics and theoretical physics, bridging classical analysis with quantum mechanics. Classical and Quantum Orthogonal Polynomials in One Variable by Mourad E. H. Ismail is a definitive reference that explores this rich subject from the perspective of special functions. Published by Cambridge University Press, this hardcover edition is an indispensable resource for graduate students, researchers, and professionals in India and around the world.

Book Overview

This volume offers a comprehensive treatment of orthogonal polynomials, covering both classical systems like Jacobi, Hermite, Laguerre, Hahn, Charlier, and Meixner polynomials, as well as modern developments such as the Askey–Wilson and Al-Salam–Chihara families. The book is unique in its inclusion of multiple orthogonal polynomials, a topic rarely found in monograph form. With updated corrections and a focus on applications in birth–death processes, integrable systems, combinatorics, and physical models, it serves as both a textbook and a research guide.

Key Highlights

• Encyclopedic coverage of classical and quantum orthogonal polynomials in one variable
• First book-length treatment of multiple orthogonal polynomials
• Modern applications in probability, physics, and combinatorics
• Open research problems and conjectures to inspire further study
• Exercises at varying difficulty for self-study and classroom use
• Thoroughly updated and corrected from the original printing

Inside the Book

The text begins with foundational concepts in special functions and orthogonal polynomials, gradually building up to advanced topics. Readers will find detailed derivations, historical notes, and connections to quantum theory. The chapter on open problems is particularly valuable for those seeking new directions in research. Each section is carefully structured to balance rigor with readability, making complex ideas accessible to Indian students pursuing pure and applied mathematics.

Key Topics

• Jacobi, Hermite, Laguerre, and other classical polynomials
• Hahn, Charlier, Meixner, and discrete orthogonal systems
• Askey–Wilson and Al-Salam–Chihara polynomials
• Multiple orthogonal polynomials and their properties
• Orthogonal polynomials in quantum mechanics and integrable systems
• Birth–death processes and stochastic models
• Combinatorial interpretations and generating functions
• Open problems and conjectures in the field

Reader Benefits

This book empowers readers with a deep understanding of orthogonal polynomials, enabling them to apply these tools in research and teaching. The exercises range from routine to challenging, helping students build confidence. Researchers will appreciate the comprehensive bibliography and the inclusion of modern results. For Indian readers, the book's clear exposition and focus on fundamental principles make it an excellent companion for advanced courses in analysis, mathematical physics, and approximation theory.

Learning Outcomes

• Master the theory and applications of classical orthogonal polynomials
• Understand the connection between orthogonal polynomials and special functions
• Gain familiarity with quantum orthogonal polynomials and their uses
• Learn to work with multiple orthogonal polynomial systems
• Develop skills to tackle open research problems in the area

Who Should Read

This book is ideal for graduate students in mathematics and physics, postdoctoral researchers, and faculty members specializing in analysis, combinatorics, or theoretical physics. It is also suitable for engineers and scientists who use orthogonal polynomials in their work. Indian students preparing for competitive exams or research careers will find it a valuable addition to their library.

About the Author

Mourad E. H. Ismail is a distinguished mathematician and professor known for his extensive contributions to special functions and orthogonal polynomials. With decades of teaching and research experience, he has authored numerous influential papers and books. His work has shaped modern understanding of the subject, and he continues to be a leading figure in the field.

About the Publisher

Cambridge University Press is a world-renowned academic publisher with a legacy of excellence in science, mathematics, and humanities. Their titles are trusted by students and researchers across India for their rigorous editorial standards and scholarly depth. This hardcover edition reflects their commitment to producing high-quality reference works.

Conclusion

Classical and Quantum Orthogonal Polynomials in One Variable is more than a book—it is a gateway to a vibrant area of mathematics. Whether you are a student seeking foundational knowledge or a researcher exploring new frontiers, this volume offers unmatched depth and breadth. Order your copy today from Bookshops.in and add this essential reference to your collection.