Guide To The Umbral Calculus, A Different Mathematical Language PDF
Preview Guide To The Umbral Calculus, A Different Mathematical Language
GGuuiiddee ttoo tthhee UUmmbbrraall CCaallccuulluuss DD AA iiffffeerreenntt MMaatthheemmaattiiccaall LLaanngguuaaggee DD B1948 Governing Asia TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk BB11994488__11--AAookkii..iinndddd 66 99//2222//22001144 44::2244::5577 PPMM GGuuiiddee ttoo tthhee UUmmbbrraall CCaallccuulluuss DD AA iiffffeerreenntt MMaatthheemmaattiiccaall LLaanngguuaaggee DD Silvia Licciardi ENEA Research Center, Frascati, Italy Giuseppe Dattoli ENEA Research Center, Frascati, Italy & University of Rome La Sapienza, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. GUIDE TO THE UMBRAL CALCULUS A Different Mathematical Language Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-125-532-8 (hardcover) ISBN 978-981-125-533-5 (ebook for institutions) ISBN 978-981-125-534-2 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12804#t=suppl Desk Editor: Nur Syarfeena Binte Mohd Fauzi Typeset by Stallion Press Email: [email protected] Printed in Singapore SSyyaarrffeeeennaa -- 1122880044 -- GGuuiiddee ttoo tthhee UUmmbbrraall CCaallccuulluuss..iinndddd 11 1188//44//22002222 1100::3377::5566 aamm May12,2022 15:12 GuidetotheUmbralCalculus-9inx6in b4660-fm pagev “A Maria Corredentrice” Silvia v May12,2022 15:12 GuidetotheUmbralCalculus-9inx6in b4660-fm pagevi “To my mother, Ada” Pino vi May12,2022 15:12 GuidetotheUmbralCalculus-9inx6in b4660-fm pagevii Contents Preface xi Acknowledgments xv Introduction xvii 1. Operator Theory and Umbral Calculus 1 1.1 From Special Functions to their Umbral Images . . 3 1.1.1 Borel Transform . . . . . . . . . . . . . . . . . . 5 1.2 Gaussian Function in Umbral Calculus . . . . . . . . 10 1.2.1 Umbral Bessel Function . . . . . . . . . . . . . 13 1.2.2 ˆb-Operator . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Principle of Permanence of the Formal Properties . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Mittag–Leffler Function: An Umbral Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2. Umbral Formulation of Hermite Polynomials 23 2.1 Theory of Special Polynomials and “Monomiality” Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Hermite Polynomial, Orthogonal Properties and Operational Formalism . . . . . . . . . . . . . . . . . . 28 2.3 An Umbral Point of View on Hermite Polynomials 33 2.4 Hermite Calculus . . . . . . . . . . . . . . . . . . . . . . 39 vii May12,2022 15:12 GuidetotheUmbralCalculus-9inx6in b4660-fm pageviii viii Guide to the Umbral Calculus: A Different Mathematical Language 2.5 Negative Derivative Operator Method and Associated Technicalities . . . . . . . . . . . . . . . . . 43 2.6 Laguerre Polynomials and Relevant Umbral Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7 Umbral Version of Laguerre and Hermite Associated Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.8 App´ell and Sheffer Polynomials: Umbral and Monomiality Points of View . . . . . . . . . . . . . . . 57 2.9 Genesis of the Monomiality Principle . . . . . . . . . 62 3. Special Polynomials and Umbral Operators 65 3.1 An Umbral Treatment of Gegenbauer, Legendre and Jacobi Polynomials . . . . . . . . . . . . . . . . . . . . . 65 3.1.1 Gegenbauer Polynomials . . . . . . . . . . . . . 71 3.1.2 Jacobi Polynomials . . . . . . . . . . . . . . . . 74 3.1.3 Legendre Polynomials . . . . . . . . . . . . . . 78 3.1.4 Generalized Forms . . . . . . . . . . . . . . . . . 81 3.2 Chebyshev, Lacunary Legendre and Legendre-type Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2.1 Umbral Methods and Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . 90 3.2.2 Legendre and Legendre-like Polynomials . . 91 4. Bessel Functions and Umbral Calculus 93 4.1 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Bessel Functions of the Second Kind. . . . . . . . . . 103 4.3 Modified Bessel Functions of the First Kind . . . . . 108 4.4 Umbra and Spherical/Stru¨ve Bessel . . . . . . . . . . 110 4.4.1 Interplay between Circular and Bessel Functions . . . . . . . . . . . . . . . . . . . . . . 114 5. Bessel Functions and Umbral Trigonometries 117 5.1 From Circular to Bessel Function . . . . . . . . . . . . 117 5.1.1 Umbral Version of the Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 117 May12,2022 15:12 GuidetotheUmbralCalculus-9inx6in b4660-fm pageix Contents ix 5.1.2 Laguerre Polynomials and Trigonometric Function . . . . . . . . . . . . . . . . . . . . . . . 120 5.2 From Laguerre to Airy Forms . . . . . . . . . . . . . . 124 5.2.1 Generalized Trigonometric Functions, Ordinary and Higher-order Bessel Functions . . . . . . . . . . . . . . . . . . . . . . 131 5.2.2 Bessel Diffusion Equations . . . . . . . . . . . 134 5.3 Pseudo-Hyperbolic Functions and Generalized Airy Diffusion Equations. . . . . . . . . . . . . . . . . . . . . 135 5.4 Generalized Trigonometric Functions and Matrix Parameterization . . . . . . . . . . . . . . . . . . . . . . 141 5.5 Evolution Equations Involving Matrices Raised to Non-Integer Exponents . . . . . . . . . . . . . . . . . . 146 5.5.1 Fractional Matrix Exponentiation . . . . . . . 150 6. Number Theory and Umbral Calculus 155 6.1 Umbral Methods and Harmonic Numbers . . . . . . 155 6.1.1 Harmonic Numbers and Generating Functions . . . . . . . . . . . . . . . . . . . . . . 155 6.1.2 Harmonic-based Functions and Differential Equations . . . . . . . . . . . . . . . . . . . . . . 159 6.1.3 Truncated Exponential Numbers . . . . . . . 164 6.2 Properties of Generalized Harmonic Numbers. . . . 164 6.3 Motzkin Numbers: an Operational Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.3.1 Motzkin Numbers and Umbral Calculus. . . 171 6.3.2 Telephone Numbers . . . . . . . . . . . . . . . . 173 7. Complements and Exercises 177 7.1 Euler Gamma, Beta Functions and Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.2 Deeper Inside Gaussians . . . . . . . . . . . . . . . . . 188 7.2.1 Hermite Calculus . . . . . . . . . . . . . . . . . 195 7.3 Generalized Transforms and Operator Methods . . 197 7.4 Umbra and Higher-Order Hermite Polynomials . . . 201
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