Spezia S. Umbral Calculus. Techniques for Pure and Applied Mathematicians 2024

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Spezia S. Umbral Calculus. Techniques for Pure and Applied Mathematicians 2024

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Textbook in PDF format 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. About the Editor Table of Contents List of Contributors List of Abbreviations Preface Introduction to Umbral Calculus and Operator Theory q-Functions and Distributions, Operational and Umbral Methods Abstract Introduction Final Comments Author Contributions Acknowledgments References 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 Hermite Polynomials in Umbral Calculus Identities Involving -Variable Hermite Polynomials Arising from Umbral Method Abstract Introduction Umbra And -variable Hermite Polynomial An Extension of the -variable Hermite Polynomials Special Cases Concluding Remarks Authors’ Contributions Acknowledgements References 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 Voigt Transform and Umbral Image Abstract Introduction Voigt Functions, Hermite Functions, and Generalized Forms Final Comments and Applications Conclusions Author Contributions References Special Polynomials in Umbral Calculus Apostol-Euler Polynomials Arising From Umbral Calculus Abstract Introduction Main Results and Applications Authors’ Contributions Acknowledgements References Barnes-type Peters Polynomial with Umbral Calculus Viewpoint Abstract Introduction Explicit Expressions Recurrence Relations Identities Authors’ Contributions Acknowledgements References 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 Sheffer Sequences of Polynomials and Their Applications Abstract Introduction Sheffer Sequences of Polynomials Authors’ Contributions Acknowledgements References Frobenius-euler Polynomials in Umbral Calculus Umbral Calculus and the FrobeniusEuler Polynomials Abstract Introduction The Frobenius-Euler Polynomials and Umbral Calculus Acknowledgment References Some Identities of Frobenius-Euler Polynomials Arising from Umbral Calculus Abstract Introduction Applications of Umbral Calculus to Frobenius-euler Polynomials Authors’ Contributions Acknowledgements References Bessel Functions In Umbral Calculus 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 Integrals of Special Functions and Umbral Methods Abstract Introduction Integrals and Non Gaussian Umbral Images Final Comments Author Contributions Acknowledgements References Number Theory and Umbral Calculus 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 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