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
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