Rusev R. Classical orthogonal polynomials and their associated functions in complex domain. - Sofia: Marin Drinov Academic Publishing House, 2005. - 278 p. - (Bulgarian Academic Monographs; 10). - Ref.: p.263-271. - Auth. ind.: p.272-274. - Sub. ind.: p.273-278. - ISBN 954-430-322-045-X  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

PREFACE ......................................................... 1 CHAPTER I. Jacobi, Laguerre and Hermite polynomials and associated functions ............................... 5 1. Pearson's differential equation .............................. 5 2. Definition of Jacobi, Laguerre and Hermite polynomials ...... 10 3. Orthogonality. Recurrence relations ......................... 12 4. Jacobi, Laguerre and Hermite associated functions. Christoffel-Darboux type formulas ........................... 24 5. Relations to hypergeometric and Weber-Hermite functions ..... 31 Exercises ...................................................... 37 Comments and references ........................................ 42 CHAPTER II. Integral representations and generating functions ......................................... 43 1. Integral representations and generating functions for Jacobi polynomials .......................................... 43 2. Integral representations and generating functions for Laguerre polynomials and associated functions ............... 46 3. Integral representations and generating functions for Hermite polynomials and associated functions ................ 53 Exercises ...................................................... 56 Comments and references ........................................ 57 CHAPTER III. Asymptotic formulas. Inequalities ................. 58 1. Asymptotic formulas for Jacobi polynomials and associated functions ................................................... 58 2. Asymptotic formulas for Hermite and Laguerre polynomials .... 67 3. Asymptotic formulas for Laguerre and Hermite associated functions ................................................... 75 4. Inequalities for Laguerre and Hermite polynomials ........... 82 5. Inequalities for Laguerre and Hermite associated functions ................................................... 86 Exercises ...................................................... 88 Comments and references ........................................ 89 CHAPTER IV. Convergence of series in Jacobi, Laguerre and Hermite systems ................................... 92 1. Series in Jacobi polynomials and associated functions ....... 92 2. Series in Laguerre polynomials and associated functions ..... 94 3. Series in Hermite polynomials and associated functions ...... 98 4. Theorems of Abelian type .................................... 99 5. Uniqueness of the representations by series in Jacobi, Laguerre and Hermite polynomials and associated functions .................................................. 103 Exercises ..................................................... 108 Comments and references ....................................... 109 CHAPTER V. Series representation of holomorphic functions by Jacobi, Laguerre and Hermite sysems ........... 1l1 1. Expansions in series of Jacobi polynomials ans associated functions .................................................. 111 2. Expansions in series of Hermite polynomials ................ 115 3. Expansions in series of Laguerre polynomials ............... 129 4. Representations by series in Laguerre and Hermite associated functions ....................................... 147 5. Holomorphic extension ...................................... 152 Exercises ..................................................... 160 Comments and references ....................................... 162 CHAPTER VI. The representation problem in terms of classical integral transforms .................... 165 1. Hankel transform and the representation by series in Laguerre polynomials ....................................... 165 2. Meijer transform and the representation by series in Laguerre associated functions .............................. 171 3. Laplace transform and the representation by series in Laguerre associated functions .............................. 177 4. Fourier transform and the representation by series in Hermite polynomials and associated functions ............... 183 5. Representation of entire functions of exponential type by series in Laguerre and Hermite polynomials .............. 185 Exercises ..................................................... 188 Comments and References ....................................... 189 CHAPTER VII. Boundary properties of series in Jacobi, Laguerre and Hermite systems ..................... 191 1. Convergence on the boundaries of convergence regions ....... 191 2. (C,δ)-summability on the boundaries of convergence regions ................................................... 205 3. Fatou type theorems ........................................ 221 Exercises ..................................................... 226 Comments and references ....................................... 228 ADDENDUM. A review on singular points and analytical continuation of series in classical orthogonal polynomials ....................................... 231 1. Singular points and analytical continuation of series in Jacobi polynomials ...................................... 231 2. Singular points and analytical continuation of series in Hermite polynomials ..................................... 236 3. Singular points and analytical continuation of series in Laguerre polynomials .................................... 241 4. Gap theorems and overconvergence ........................... 244 Comments and references ....................................... 249 APPENDIX. A short survey on special functions ............... 252 1. Gamma-function ............................................. 252 2. Bessel functions ........................................... 254 3. Hypergeometric functions ................................... 257 4. Weber-Hermite functions .................................... 261 Comments and references ....................................... 262 REFERENCES .................................................... 263 AUTHOR INDEX .................................................. 272 SUBJECT INDEX ................................................. 275