ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS
Preface page xiii 1 The Gamma and Beta Functions 1 1.1 The Gamma and Beta Integrals and Functions 2 1.2 The Euler Reflection Formula 9 1.3 The Hurwitz and Riemann Zeta Functions 15 1.4 Stirling's Asymptotic Formula 18 1.5 Gauss's Multiplication Formula for T(mx) 22 1.6 Integral Representations for Log T(x) and xjr (x) 26 1.7 Kummer's Fourier Expansion of Log F (x) 29 1.8 Integrals of Dirichlet and Volumes of Ellipsoids 32 1.9 The Bohr-Mollerup Theorem 34 1.10 Gauss and Jacobi Sums 36 1.11 A Probabilistic Evaluation of the Beta Function. 43 1.12 The /? : adic Gamma Function 44 46 2 The Hypergeometric Functions \ 61 2.1 The Hypergeometric Series 61 2.2 Euler's Integral Representation 65 2.3 The Hypergeometric Equation 73 2.4 The Barnes Integral for the Hypergeometric Function 85 2.5 Contiguous Relations 94 2.6 Dilogarithms 102 2.7 Binomial Sums 107 2.8 Dougall's Bilateral Sum 109 2.9 Fractional Integration by Parts and Hypergeometric Integrals 111 114
viii 3 Hypergeometric Transformations and Identities 124 3.1 Quadratic Transformations 125 3.2 The Arithmetic-Geometric Mean and Elliptic Integrals 132 3.3 Transformations of Balanced Series 140 3.4 Whipple's Transformation 143 3.5 Dougall's Formula and Hypergeometric Identities 147 3.6 Integral Analogs of Hypergeometric Sums 150 3.7 Contiguous Relations 154 3.8 The Wilson Polynomials 157 3.9 Quadratic Transformations - Riemann's View 160 3.10 Indefinite Hypergeometric Summation 163 3.11 The W-Z Method 166 3.12 Contiguous Relations and Summation Methods 174 176 4 Bessel Functions and Confluent Hypergeometric Functions 187 4.1 The Confluent Hypergeometric Equation 188 4.2 Barnes's Integral for i F] 192 4.3 Whittaker Functions 195 4.4 Examples of i Fi and Whittaker Functions 196 4.5 Bessel's Equation and Bessel Functions 199 4.6 Recurrence Relations 202 4.7 Integral Representations of Bessel Functions 203 4.8 Asymptotic Expansions 209 4.9 Fourier Transforms and Bessel Functions 210 4.10 Addition Theorems 213 4.11- Integrals of Bessel Functions 216 4.12 The Modified Bessel Functions 222 4.13 Nicholson's Integral 223 4.14 Zeros of Bessel Functions 225 4.15 Monotonicity Properties of Bessel Functions 229 4.16 Zero-Free Regions for ifi Functions 231 234 5 Orthogonal Polynomials 240 5.1 Chebyshev Polynomials 240 5.2 Recurrence 244 5.3 Gauss Quadrature 248 5.4 Zeros of Orthogonal Polynomials 253 5.5 Continued Fractions 256
5.6 Kernel Polynomials 259 5.7 Parseval's Formula 263 5.8 The Moment-Generating Function 266 269 6 Special Orthogonal Polynomials 277 6.1 Hermite Polynomials 278 6.2 Laguerre Polynomials 282 6.3 Jacobi Polynomials and Gram Determinants 293 6.4 Generating Functions for Jacobi Polynomials 297 6.5 Completeness of Orthogonal Polynomials 306 6.6 Asymptotic Behavior of P^a- P) {x) for Large n 310 6.7 Integral Representations of Jacobi Polynomials 313 6.8 Linearization of Products of Orthogonal Polynomials 316 6.9 Matching Polynomials 323 6.10 The Hypergeometric Orthogonal Polynomials 330 6.11 An Extension of the Ultraspherical Polynomials 334 339 7 Topics in Orthogonal Polynomials 355 7.1 Connection Coefficients 356 7.2 Rational Functions with Positive Power Series Coefficients 363 7.3 Positive Polynomial Sums from Quadrature and Vietoris's Inequality 371 7.4 Positive Polynomial Sums and the Bieberback Conjecture 381 7.5 A Theorem of Turan 384 7.6 Positive Summability of Ultraspherical Polynomials 388 7.7 The Irrationality of? (3) 391 395 8 The Selberg Integral and Its Applications 401 8.1 Selberg's and Aomoto's Integrals 402 8.2 Aomoto's Proof of Selberg's Formula 402 8.3 Extensions of Aomoto's Integral Formula 407 8.4 Anderson's Proof of Selberg's Formula 411 8.5 A Problem of Stieltjes and the Discriminant of a Jacobi Polynomial 415 8.6 Siegel's Inequality 419 8.7 The Stieltjes Problem on the Unit Circle 425 8.8 Constant-Term Identities 426 8.9 Nearly Poised 3F 2 Identities 428 8.10 The Hasse-Davenport Relation 430 ix
x 8.11 A Finite-Field Analog of Selberg's Integral 434 439 9 Spherical Harmonics 445 9.1 Harmonic Polynomials - 445 9.2 The Laplace Equation in Three Dimensions 447 9.3 Dimension of the Space of Harmonic Polynomials of Degree k 449 9.4 Orthogonality of Harmonic Polynomials 451 9.5 Action of an Orthogonal Matrix 452 9.6 The Addition Theorem 454 9.7 The Funk-Hecke Formula 458 9.8 The Addition Theorem for Ultraspherical Polynomials 459 9.9 The Poisson Kernel and Dirichlet Problem 463 9.10 Fourier Transforms 464 9.11 Finite-Dimensional Representations of Compact Groups 466 9.12 The Group SU(2) 469 9.13 Representations of SU (2) 471 9.14 Jacobi Polynomials as Matrix Entries 473 9.15 An Addition Theorem 474 9.16 Relation of SI/ (2) to the Rotation Group S O (3) 476 478 10 Introduction to ^-Series 481 10.1 The ^-Integral 485 10.2 The ^-Binomial Theorem. 487 10.3 The q -Gamma Function 493 10.4* The Triple Product Identity 496 10.5 Ramanujan's Summation Formula 501 10.6 Representations of Numbers as Sums of Squares 506 10.7 Elliptic and Theta Functions 508 10.8 q -Beta Integrals 513 10.9 Basic Hypergeometric Series 520 10.10 Basic Hypergeometric Identities 523 10.11 g-ultraspherical Polynomials, 527 10.12 Mellin Transforms 532 542 11 Partitions 553 11.1 Background on Partitions 553 11.2 Partition Analysis 555 11.3 A Library for the Partition Analysis Algorithm 557
12 B D E 11.4 Generating Functions 11.5 Some Results on Partitions 11.6 Graphical Methods 11.7 Congruence Properties of Partitions Bailey Chains 12.1 Rogers's Second Proof of the Rogers-Ramanujan Identities 12.2 Bailey's Lemma 12.3 Watson's Transformation Formula 12.4 Other Applications Infinite Products A. 1 Infinite Products Summability and Fractional Integration B.I Abel and Cesaro Means B.2 The Cesaro Means (C, a) B.3 Fractional Integrals B.4 Historical Remarks Asymptotic Expansions C.I Asymptotic Expansion C.2 Properties of Asymptotic Expansions C.3 Watson's Lemma C.4 The Ratio of Two Gamma Functions Euler-Maclaurin Summation Formula D.I Introduction D.2 The Euler-Maclaurin Formula D.3 Applications D.4 The Poisson Summation Formula Lagrange Inversion Formula E. 1 Reversion of Series E.2 A Basic Lemma E.3 Lambert's Identity E.4 Whipple's Transformation 559 563 565 569 573 577 577 582 586 589 590 595 595 597 599 599 602 604 605 607 611 611 612 614 615 616 617 617 619 621 623 627 629 629 630 631 632 634
xii F Series Solutions of Differential Equations 637 F.I Ordinary Points 637 F.2 Singular Points 638 F.3 Regular Singular Points 639 Bibliography 641 Index 655 Subject Index 659 Symbol Index 661