Special Functions and Generalized Sturm-Liouville Problems

Preface......Page 6
Contents......Page 10
1.1 Introduction......Page 13
1.2 Pearson Distribution Families and Gauss Hypergeometric Functions......Page 14
1.3 Six Sequences of Hypergeometric Orthogonal Polynomials......Page 24
1.3.2 Laguerre Polynomials......Page 25
1.3.3 Hermite Polynomials......Page 26
1.3.4 First Finite Sequence of Hypergeometric Orthogonal Polynomials......Page 27
1.3.5 Second Finite Sequence of Hypergeometric Orthogonal Polynomials......Page 29
1.3.6 Third Finite Sequence of Hypergeometric Orthogonal Polynomials......Page 32
1.4 A Generic Polynomial Solution for the Classical Hypergeometric Differential Equation......Page 37
1.4.1 Six Special Cases of the Generic Polynomials n(x;a,b,c,d,e)......Page 48
1.4.2 How to Find the  Initial Vector If a Special Case of the Main Weight Function Is Given?......Page 52
1.5 Fourier Transforms of Finite Sequences of Hypergeometric Orthogonal Polynomials......Page 54
1.6 A  Symmetric Generalization of Sturm–Liouville Problems......Page 59
1.7 A Basic Class of Symmetric Orthogonal Polynomials......Page 61
1.7.1 A Direct Relationship Between  n(x;a,b,c,d,e)  and n(x;p,q,r,s)......Page 66
1.7.2 Four Special Cases of the Symmetric Polynomials  Sn(x;p,q,r,s)......Page 68
1.7.3 A Unified Approach for the Classification of  n(x;p,q,r,s)......Page 78
1.8 Fourier Transforms of Symmetric Orthogonal Polynomials......Page 81
1.8.1 Fourier Transform of Generalized Ultraspherical Polynomials......Page 82
1.8.2 Fourier Transform of Generalized Hermite Polynomials......Page 84
1.8.3 Fourier Transform of the First Finite Sequence of Symmetric Orthogonal Polynomials......Page 88
1.8.4 Fourier Transform of the Second Finite Sequence of Symmetric Orthogonal Polynomials......Page 92
1.9 A Class of Symmetric Orthogonal Functions......Page 95
1.9.1 Four Special Cases of  Sn(θ)(x;p,q,r,s)......Page 100
1.10 An Extension of  Sn(θ)(x;p,q,r,s)......Page 105
1.10.1 Four Special Cases of  Sn(α,β)(x;p,q,r,s)......Page 111
1.11 A Generalization of Fourier Trigonometric Series......Page 115
1.11.1 A Generalization of Trigonometric Orthogonal Sequences......Page 116
1.11.2 Application to Function Expansion Theory......Page 120
1.12 Another Extension for Trigonometric Orthogonal Sequences......Page 124
1.13.1 Incomplete Symmetric Orthogonal Polynomials of Jacobi Type......Page 130
1.13.2 Incomplete Symmetric Orthogonal Polynomials of Laguerre Type......Page 136
1.14 A Class of Hypergeometric-Type Orthogonal Functions......Page 140
1.15 Application of Zero Eigenvalue in Solving Some Sturm–Liouville Problems......Page 146
1.15.1 A Relationship Between the Chebyshev Polynomials of the Third and Fourth Kinds and the Associated Legendre Differential Equation......Page 156
Further Reading......Page 164
2.1 Introduction......Page 168
2.2 A Finite Sequence of  Hahn-Type Orthogonal Polynomials......Page 174
2.3 Classical Symmetric Orthogonal Polynomials of a Discrete Variable......Page 179
2.3.1 Classification......Page 182
2.4 A Symmetric Generalization of Sturm–Liouville Problems in Discrete Spaces......Page 189
2.4.1 Some Illustrative Examples......Page 194
2.5 A Basic Class of Symmetric Orthogonal Polynomials of a Discrete Variable......Page 208
2.5.1 Two Infinite Types of Sn*(x;a,b,c,d)......Page 217
2.5.2 Moments of the Two Introduced Infinite Sequences......Page 227
2.5.3 A Special Case of Sn*(x;a,b,c,d) Generating All Classical Symmetric Orthogonal Polynomials of a Discrete Variable......Page 228
2.5.4 Two Finite Types of  Sn*(x;a,b,c,d)......Page 230
Further Reading......Page 240
3.1 Introduction......Page 243
3.2.1 The q-Shifted Factorial......Page 246
3.2.2 q-Hypergeometric Series......Page 247
3.2.3 q-Binomial Coefficients and the q-Binomial Theorem......Page 248
3.2.5 q-Analogues of Some Special Functions......Page 249
3.2.6 q-Difference Operators......Page 250
3.2.7 q-Integral Operators......Page 252
3.2.8 An Analytical Solution for the  q -Pearson DifferenceEquation......Page 255
3.2.9 Difference Equations of  q -Hypergeometric Series......Page 256
3.2.10 q-Analogues of Jacobi Polynomials......Page 258
3.2.11 q-Analogues of Laguerre Polynomials......Page 260
3.2.12 q-Analogue of Hermite Polynomials......Page 263
3.2.13 A Biorthogonal Exponential Sequence......Page 264
Some Remarks on Theorem 3.2......Page 269
3.3 Three Finite Classes of q-Orthogonal Polynomials......Page 272
3.3.1 First Finite Sequence of q-Orthogonal Polynomials Corresponding to the Inverse Gamma Distribution......Page 273
3.3.2 Second Finite Sequence of q-Orthogonal Polynomials Corresponding to the Fisher Distribution......Page 280
3.3.3 Third Finite Sequence of q-Orthogonal Polynomials Corresponding to Student's t-Distribution......Page 285
3.3.4 A Characterization of Three Introduced Finite Sequences......Page 291
Comparison with the Third Finite Sequence......Page 292
3.4 A Symmetric Generalization of Sturm–Liouville Problems in q-Difference Spaces......Page 293
3.4.1 Some Illustrative Examples......Page 296
3.5 A Basic Class of Symmetric q-Orthogonal Polynomials with Four Free Parameters......Page 300
3.5.1 Two Finite Sequences Based on Ramanujan's Identity......Page 308
References......Page 316
Further Reading......Page 317
Index......Page 320