Generalized Mathieu Series

The Mathieu series is a functional series introduced by Émile Léonard Mathieu for the purposes of his research on the elasticity of solid bodies. Bounds for this series are needed for solving biharmonic equations in a rectangular domain. In addition to Tomovski and his coauthors, Pogany, Cerone, H. M. Srivastava, J. Choi, etc. are some of the known authors who published results concerning the Mathieu series, its generalizations and their alternating variants. Applications of these results are given in classical, harmonic and numerical analysis, analytical number theory, special functions, mathematical physics, probability, quantum field theory, quantum physics, etc. Integral representations, analytical inequalities, asymptotic expansions and behaviors of some classes of Mathieu series are presented in this book. A systematic study of probability density functions and probability distributions associated with the Mathieu series, its generalizations and Planck's distributionis also presented. The book is addressed at graduate and PhD students and researchers in mathematics and physics who are interested in special functions, inequalities and probability distributions.

Gives an overview of the research based on Mathieu series Provides cases of asymptotic expansions of Mathieu series Addressed at graduate and PhD students and researchers in mathematics and physics

Auteur

Stefan Gerhold is a full professor in the research unit of financial and actuarial mathematics, TU Wien, Austria. He holds a PhD from Johannes Kepler University, Linz and has published approximately 50 papers in peer-reviewed journals. His main research interests are mathematical finance and probability theory, with a focus on asymptotic methods, as well as special functions and symbolic computation.

Delco LeSkovski is an assistant professor at International Balkan University, Republic of North Macedonia. His research interests include mathematical analysis, special functions, Mathieu series, probability and statistics and differential equations. He is an author or coauthor of several articles in mathematical peer-reviewed journals.

Zivorad Tomovski has held the position of full professor since 2010 at the Ss. Cyril and Methodius University of Skopje, North Macedonia. He held visiting researcher positions at many prestigious universities and research institutes across Europe. Since 2021, he has been an associate professor of applied analysis and probability and statistics at the University of Ostrava, Czech Republic. He has published 1 Springer book and over 100 international papers in the areas of pure and applied mathematics.

Texte du rabat

The Mathieu series is a functional series introduced by Émile Léonard Mathieu for the purposes of his research on the elasticity of solid bodies. Bounds for this series are needed for solving biharmonic equations in a rectangular domain. In addition to Tomovski and his coauthors, Pogany, Cerone, H. M. Srivastava, J. Choi, etc. are some of the known authors who published results concerning the Mathieu series, its generalizations and their alternating variants. Applications of these results are given in classical, harmonic and numerical analysis, analytical number theory, special functions, mathematical physics, probability, quantum field theory, quantum physics, etc. Integral representations, analytical inequalities, asymptotic expansions and behaviors of some classes of Mathieu series are presented in this book. A systematic study of probability density functions and probability distributions associated with the Mathieu series, its generalizations and Planck s distribution is also presented. The book is addressed at graduate and PhD students and researchers in mathematics and physics who are interested in special functions, inequalities and probability distributions.

Résumé
"The authors discuss in great detail integral representations, series representations and inequalities for various classes of generalized Mathieu series. Asymptotic expansions of these Mathieu series, ... relations of generalized Mathieu series and probability theory are the main topics related to the generalized Mathieu series discussed in this well written book. The book is of interest for researchers of classical analysis working in the field of special functions or inequalities for functions defined on the real axis." ( Árpád Baricz, Mathematical Reviews, December, 2022)

Contenu
1 Introduction.- 2 Generalized Mathieu Series.- 3 Mean Convergence of Fourier-Mathieu Series.- 4 Estimates for Multiple Generalized Mathieu Series.- 5 Asymptotic Expansions of Mathieu Series.- 6 Two-Sided Inequalities for the Butzer-Flocke-Hauss Complete Omega Function.- 7 Probability Distributions Associated with Mathieu Series.- 8 Conclusion.- Appendix A: Some special functions and their properties. <p