Theory and Application of Special Functions : Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, the University of Wisconsin-Madison, March 31-April 2 1975.

Front Cover -- Theory and Application of Special Functions -- Copyright Page -- Table of Contents -- Foreword -- Preface -- Chapter 1. Computational Methods in Special Functions-A Survey -- Introduction -- 1. Methods based on preliminary approximation -- 2. Methods based on linear recurrence relations -- 3. Nonlinear recurrence algorithms for elliptic integrals and elliptic functions -- 4. Computer software for special functions -- REFERENCES -- Chapter 2. Unsolved Problems in the Asymptotic Estimation of Special Functions -- Abstract -- 1. INTRODUCTION -- PART I. THE TOOLS OF ASYMPTOTIC ANALYSIS -- 2. INTEGRALS -- 3. SUMS AND SEQUENCES -- 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS -- PART II. ASYMPTOTIC ESTIMATES OF THE SPECIAL FUNCTIONS -- 5. FUNCTIONS OF ONE OR TWO VARIABLES -- 6. FUNCTIONS OF THREE VARIABLES -- 7. FUNCTIONS OF FOUR OR MORE VARIABLES -- ACKNOWLEDGMENTS -- REFERENCES -- Chapter 3. Periodic Bernoulli Numbers, Summation Formulas and Applications -- 1. Introduction. -- 2. Periodic Bernoulli numbers and polynomials -- 3. The periodic Poisson and periodic Euler-Maclaurin summation -- 4. The distribution of quadratic residues -- 5. Power sums and cotangent sums -- 6. Gauss sums -- 7. Functional equations -- 8. A trigonometric series of Hardy and Littlewood -- 9. Infinite series of ordinary Bessel functions -- 10. Infinite series of modified Bessel functions -- 11. Entries from Ramanujan's Notebooks and kindred formulae -- REFERENCES -- Chapter 4. Problems and Prospects for Basic Hypergeometric Functions -- 1. Introduction -- 2. Partitions identities -- 3. Identities for Multiple Hypergeometric Series -- 4. Basic Appell and Lauricella Series -- 5. MacMahon's Master Theorem and the Dyson Conjecture -- 6. Saalschützian Series and Inversion Theorems -- 7. Conclusion. -- REFERENCES.

Chapter 5. An Introduction to Association Schemes and Coding Theory -- ABSTRACT -- 1 INTRODUCTION -- 2 Error-Correcting Codes -- 3 Association Schemes -- 4 The Hamming Association Scheme -- 5 The Johnson Association Scheme -- 6 Association Schemes Obtained from Graphs and Other Sources -- 7 The Linear Programming Bound -- 8 Properties of Perfect Codes -- REFERENCES -- Chapter 6. Linear Growth Models with Many Types and Multidimensional Hahn Polynomials -- 1. Multi-allele Moran mutation models -- 2. Representation of P(t). -- 3. Relation with multi-dimensional linear growth -- 4. The case r = 2 and the Hahn polynomials -- 5. Moran model with r types. -- 6. Linear growth model with r types -- 7. The eigenfunctions when -- REFERENCES -- Chapter 7. Orthogonal Polynomials Revisited -- I. Introduction -- II. Polynomials on the Real Axis -- III. Applications -- IV. Polynomials on the Unit Circle -- V. Conclusion -- FOOTNOTES -- Chapter 8. Symmetry, Separation of Variables, and Special Functions -- REFERENCES -- Chapter 9. Nicholson-Type Integrals for Products of Gegenbauer Functions and Related Topics -- ABSTRACT -- 1. INTRODUCTION -- 2. DERIVATION OF A NICHOLSON-TYPE FORMULA FOR GEGENBAUER FUNCTIONS -- 3. SOME APPLICATIONS FOR GEGENBAUER FUNCTIONS -- 4. DEDUCTIONS FOR OTHER FUNCTIONS -- 5. CONCLUSIONS -- FOOTNOTES -- REFERENCES -- Chapter 10. Positivity and Special Functions -- 1. Introduction -- 2. Background material. -- 3. Computation. -- 4. Recurrence relations. -- 5. Maximum principles. -- 6. Estimation. -- 7. Absolutely monotonie and completely monotonie functions -- 8. Sums of squares. -- 9. Additional comments. -- REFERENCES -- Chapter 11. Two-Variable Analogues of the Classical Orthogonal Polynomials -- 1. Introduction -- 2. Jacobi polynomials -- 3. Methods of constructing two-variable analogues of the Jacobipolynomials.

4. Differential recurrence relations and other analytic properties -- 5. Orthogonal polynomials in two variables as spherical functions -- REFERENCES -- Chapter 12. Special Functions of Matrix and Single Argument in Statistics -- FOREWORD -- INTRODUCTION -- I. Zonal Polynomials of the Real Positive Definite Symmetric Matrices -- II. Orthogonal Polynomials of Matrix Argument -- III. Differential Equations for Hypergeometric Functions of Matrix Argument -- IV. Probability Functions of Ordered Roots -- REFERENCES -- Chapter 13. Some Properties of Determinants of Orthogonal Polynomials -- Introduction -- 1. Recurrence formulas -- 2. CHRISTQFFEL-DAJRBOUX FORMULA AND WRONSKIAN IDENTITIES. -- 3. Further Properties -- REFERENCES -- Index.

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