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.
Author(s): Živorad Tomovski, Delčo Leškovski, Stefan Gerhold
Publisher: Springer
Year: 2021
Language: English
Pages: 175
City: Cham
Preface
About This Book
Contents
About the Authors
1 Generalized Mathieu Series, Associated Integral Representations and Convergence
1.1 Mathieu Series and First Generalizations
1.2 A Family of Multi-parameter Mathieu Series
1.3 (p,q)-Mathieu Type Power Series
1.4 Turán Type Inequalities
1.5 (a,λ)-Mathieu Series
1.6 Trigonometric Mathieu Series
1.7 Mathieu Series Associated with the Three-Parameter Mittag-Leffler Function
1.8 A Class of Harmonic Mathieu Series
2 Mean Convergence of Fourier-Mathieu Series
3 Estimates for Multiple Generalized Mathieu Series
3.1 Multiple Generalized Mathieu Series
3.2 Hardy-Hilbert Type Inequalities
3.3 Inequalities for Multiple Generalized Mathieu Series
4 Asymptotic Expansions of Mathieu Series
4.1 Overview
4.2 Power-Logarithmic Sequences
4.3 Factorial Sequences: The Associated Dirichlet Series
4.4 Factorial Sequences: Proofs
4.5 An Alternative Proof of the Expansion of Sµ(r)
4.6 Miscellaneous
5 Two-Sided Inequalities for the Butzer-Flocke-Hauss Complete Omega Function
5.1 Butzer-Flocke-Hauss Complete Omega Function
5.2 Čaplygin Type Differential Inequality
5.3 Bilateral Bounds
5.3.1 Two-Sided Inequalities Associated with the Class R
5.3.2 Two-Sided Inequalities for the Class A
5.3.3 Two-Sided Inequalities Associated with the Class O
5.3.4 Two-Sided Inequalities Associated with the Explicit Bounds on widetildeS(x)
5.3.5 Efficiency of the Bounds
6 Probability Distributions Associated with Mathieu Series
6.1 Mathieu Distributions
6.2 Probability Distributions Associated with a Generalization of Planck's Law
Appendix Conclusion
Appendix Appendix A Some Special Functions and Their Properties
Gamma and Beta Functions
Asymptotic inversion of the gamma function
Psi function and harmonic numbers
The General Hurwitz-Lerch zeta function and some generalizations
Bessel Function
Mittag-Leffler Functions
Appendix References
Index
