Book:George E. Andrews/Special Functions

Contents

 * Preface


 * CHAPTER 1: The Gamma and Beta Functions
 * $\S 1$. The Gamma and Beta Integrals and Functions
 * $\S 2$. The Euler Reflection Formula
 * $\S 3$. The Hurwitz and Riemann Zeta Functions
 * $\S 4$. Stirling's Asymptotic Formula
 * $\S 5$. Gauss's Multiplication Formula for $\map \Gamma {mx}$
 * $\S 6$. Integral Representation for $\map \Log {\map \Gamma x}$ and $\map \psi x$
 * $\S 7$. Kummer's Fourier Expansion of $\map \Log {\map \Gamma x}$
 * $\S 8$. Integrals of Dirichlet and Volumes of Ellipsoids
 * $\S 9$. The Bohr-Mollerup Theorem
 * $\S 10$. Gauss and Jacobi Sums
 * $\S 11$. A Probabilistic Evaluation of the Beta Function
 * $\S 12$. The p-adic Gamma Function
 * Exercises


 * CHAPTER 2: The Hypergeometric Functions
 * $\S 1$. The Hypergeometric Series
 * $\S 2$. Euler's Integral Representation
 * $\S 3$. The Hypergeometric Equation
 * $\S 4.$ The Barnes Integral for the Hypergeometric Function
 * $\S 5$. Contiguous Relations
 * $\S 6$. Dilogarithms
 * $\S 7$. Binomial Sums
 * $\S 8$. Dougall's Bilateral Sum
 * $\S 9$. Fractional Integration by Parts and Hypergeometric Integrals
 * Exercises


 * CHAPTER 3: Hypergeometric Transformations and Identities
 * $\S 1$. Quadratic Transformations
 * $\S 2$. The Arithmetic-Geometric Mean and Elliptic Integrals
 * $\S 3$. Transformations of Balanced Series
 * $\S 4.$ Whipple's Transformation
 * $\S 5$. Dougall's Formula and Hypergeometric Identities
 * $\S 6$. Integral Analogs of Hypergeometric Sums
 * $\S 7$. Contiguous Relations
 * $\S 8$. The Wilson Polynomials
 * $\S 9$. Quadratic Transformations - Riemann's View
 * $\S 10$. Indefinite Hypergeometric Summation
 * $\S 11$. The W-Z Method
 * $\S 12$. Contiguous Relations and Summation Methods
 * Exercises

More to complete later...(10+ additional chapters)


 * Bibliography
 * Index
 * Subject Index
 * Symbol Index