Guide To The Umbral Calculus, A Different Mathematical Language
- Length: 294 pages
- Edition: 1
- Language: English
- Publisher: WSPC
- Publication Date: 2022-06-16
- ISBN-10: 9811255326
- ISBN-13: 9789811255328
- Sales Rank: #8810800 (See Top 100 Books)
This book covers different aspects of umbral calculus and of its more recent developments. It discusses the technical details in depth, including its relevant applications. The book has therefore manyfold scopes to introduce a mathematical tool, not widespread known as it should be; to present a complete account of the relevant capabilities through the use of different examples of applications; to provide a formal bridge between different fields of research in pure and applied.
Cover Page Title Page Copyright Page Dedication Contents Preface Acknowledgments Introduction 1. Operator Theory and Umbral Calculus 1.1 From Special Functions to their Umbral Images 1.1.1 Borel Transform 1.2 Gaussian Function in Umbral Calculus 1.2.1 Umbral Bessel Function 1.2.2 b̂-Operator 1.2.3 Principle of Permanence of the Formal Properties 1.3 Mittag–Leffler Function: An Umbral Point of View 2. Umbral Formulation of Hermite Polynomials 2.1 Theory of Special Polynomials and “Monomiality” Principle 2.2 Hermite Polynomial, Orthogonal Properties and Operational Formalism 2.3 An Umbral Point of View on Hermite Polynomials 2.4 Hermite Calculus 2.5 Negative Derivative Operator Method and Associated Technicalities 2.6 Laguerre Polynomials and Relevant Umbral Forms 2.7 Umbral Version of Laguerre and Hermite Associated Polynomials 2.8 Appéll and Sheffer Polynomials: Umbral and Monomiality Points of View 2.9 Genesis of the Monomiality Principle 3. Special Polynomials and Umbral Operators 3.1 An Umbral Treatment of Gegenbauer, Legendre and Jacobi Polynomials 3.1.1 Gegenbauer Polynomials 3.1.2 Jacobi Polynomials 3.1.3 Legendre Polynomials 3.1.4 Generalized Forms 3.2 Chebyshev, Lacunary Legendre and Legendre-type Polynomials 3.2.1 Umbral Methods and Chebyshev Polynomials 3.2.2 Legendre and Legendre-like Polynomials 4. Bessel Functions and Umbral Calculus 4.1 Bessel Functions 4.2 Bessel Functions of the Second Kind 4.3 Modified Bessel Functions of the First Kind 4.4 Umbra and Spherical/Strüve Bessel 4.4.1 Interplay between Circular and Bessel Functions 5. Bessel Functions and Umbral Trigonometries 5.1 From Circular to Bessel Function 5.1.1 Umbral Version of the Trigonometric Functions 5.1.2 Laguerre Polynomials and Trigonometric Function 5.2 From Laguerre to Airy Forms 5.2.1 Generalized Trigonometric Functions, Ordinary and Higher-order Bessel Functions 5.2.2 Bessel Diffusion Equations 5.3 Pseudo-Hyperbolic Functions and Generalized Airy Diffusion Equations 5.4 Generalized Trigonometric Functions and Matrix Parameterization 5.5 Evolution Equations Involving Matrices Raised to Non-Integer Exponents 5.5.1 Fractional Matrix Exponentiation 6. Number Theory and Umbral Calculus 6.1 Umbral Methods and Harmonic Numbers 6.1.1 Harmonic Numbers and Generating Functions 6.1.2 Harmonic-based Functions and Differential Equations 6.1.3 Truncated Exponential Numbers 6.2 Properties of Generalized Harmonic Numbers 6.3 Motzkin Numbers: an Operational Point of View 6.3.1 Motzkin Numbers and Umbral Calculus 6.3.2 Telephone Numbers 7. Complements and Exercises 7.1 Euler Gamma, Beta Functions and Related Topics 7.2 Deeper Inside Gaussians 7.2.1 Hermite Calculus 7.3 Generalized Transforms and Operator Methods 7.4 Umbra and Higher-Order Hermite Polynomials 7.5 Operator Ordering and Laguerre Umbral Operators 7.6 Ramanujan Master Theorem 7.7 Products of Bessel Functions and Associated Polynomials 7.7.1 lr{ν}(k) Polynomials 7.8 Mittag–Leffler Functions and Fractional Calculus 7.8.1 Mittag–Leffler–Hermite Polynomials 7.9 Voigt Functions 7.10 Applications to Physics 7.10.1 Fractional Schrödinger Equation 7.10.2 Pearcey Integral in Optics and Evolutive Problems 7.10.3 Free Electron Laser 7.10.4 Bethe–Salpeter Equation and Matrix Environment Conclusions Bibliography Index
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