Umbral Calculus: Techniques for Pure and Applied Mathematicians

Since 1850, mathematicians have successfully applied umbral calculus in many fields of mathematics and physics. The success of umbral calculus is due to the possibility of using techniques that have simplified the technicalities of calculations, which are usually wearisome when performed with conventional methods. Umbral Calculus: Techniques for Pure and Applied Mathematics book provides the theoretical basis and many examples of umbral calculus, including operator theory, Hermite, Frobenius-Euler, and other special polynomials, Bessel functions, and at the end, results concerning number theory within umbral calculus viewpoint. Cover Half Title Page Title Page Copyright Declaration About the Editor Table of Contents List of Contributors List of Abbreviations Preface Section 1: Introduction to Umbral Calculus and Operator Theory Chapter 1: q-Functions and Distributions, Operational and Umbral Methods Abstract Introduction Final Comments Author Contributions Acknowledgments References Chapter 2: Dual Numbers and Operational Umbral Methods Abstract Introduction Higher-order Dual Numbers Umbral-type Methods and Dual Numbers Dual Numbers and Solution of Heat- and Schrödinger-Type Equations Weyl Formula and Modified Hermite Polynomials Final Comments Author Contributions Acknowledgments References Section 2: Hermite Polynomials in Umbral Calculus Chapter 3: Identities Involving 3-Variable Hermite Polynomials Arising from Umbral Method Abstract Introduction Umbra And 3-variable Hermite Polynomial An Extension of the 3-variable Hermite Polynomials Special Cases Concluding Remarks Authors’ Contributions Acknowledgements References Chapter 4: Some New Identities of Bernoulli, Euler and Hermite Polynomials Arising From Umbral Calculus Abstract Introduction Some Identities of Several Special Polynomials Authors’ Contributions Acknowledgements References Chapter 5: Voigt Transform and Umbral Image Abstract Introduction Voigt Functions, Hermite Functions, and Generalized Forms Final Comments and Applications Conclusions Author Contributions References Section 3: Special Polynomials in Umbral Calculus Chapter 4: Apostol-Euler Polynomials Arising From Umbral Calculus Abstract Introduction Main Results and Applications Authors’ Contributions Acknowledgements References Chapter 5: Barnes-type Peters Polynomial with Umbral Calculus Viewpoint Abstract Introduction Explicit Expressions Recurrence Relations Identities Authors’ Contributions Acknowledgements References Chapter 6: Representation by Degenerate Genocchi Polynomials Abstract Introduction And Preliminaries Review of Umbral Calculus Representation by Degenerate Genocchi Polynomials Representation by Higher-order Degenerate Genocchi Polynomials Examples Conclusion and Future Work Acknowledgments References Chapter 7: Sheffer Sequences of Polynomials and Their Applications Abstract Introduction Sheffer Sequences of Polynomials Authors’ Contributions Acknowledgements References Section 4: Frobenius-euler Polynomials in Umbral Calculus Chapter 10: Umbral Calculus and the FrobeniusEuler Polynomials Abstract Introduction The Frobenius-Euler Polynomials and Umbral Calculus Acknowledgment References Chapter 11: Some Identities of Frobenius-Euler Polynomials Arising from Umbral Calculus Abstract Introduction Applications of Umbral Calculus to Frobenius-euler Polynomials Authors’ Contributions Acknowledgements References Section 5: Bessel Functions In Umbral Calculus Chapter 12: A Determinant Expression for the Generalized Bessel Polynomials Abstract Introduction Exponential Riordan Array Bessel Polynomials and Bessel Matrices Determinant Formulae for Bessel Polynomials Acknowledgments References Chapter 13: Integrals of Special Functions and Umbral Methods Abstract Introduction Integrals and Non Gaussian Umbral Images Final Comments Author Contributions Acknowledgements References Section 6: Number Theory and Umbral Calculus Chapter 14: Poly-Cauchy Numbers and Polynomials of the Second Kind with Umbral Calculus Viewpoint Abstract Introduction Umbral Calculus Poly-cauchy Numbers and Polynomials of the Second Kind Authors’ Contributions Acknowledgements References Chapter 15: Extended R-Central Bell Polynopmials with Umbral Calculus Viewpoint Abstract Introduction and Preliminaries Quick Review of Umbral Calculus Main Results Conclusions Authors’ Contributions Acknowledgements References Index Back Cover