Extended Hypergeometric Functions and Orthogonal Polynomials presents a comprehensive and accessible resource for researchers and graduate students interested in exploring the rich connections between extended hypergeometric functions, orthogonal polynomials, and multivariable polynomials. Integrating all three fields and their applications in Maple, Mathematica, and MATLAB, this book fosters interdisciplinary understanding and inspires new avenues of research in mathematics, engineering, physics, and computer science. It also provides a glimpse into future research directions in these areas, including potential applications in emerging fields of applied mathematics and interdisciplinary collaborations. Each chapter begins with an introduction, includes sections on theory, followed by sections on applications, and ends with exercises, problems, references and suggested readings.
1. Introduction - p. xvii 2. Chapter 1: The generalized k-hypergeometric function as the Meijer G-function - p. 1 3. Chapter 2: Matrix analog of a new class of hypergeometric functions - p. 13 4. Chapter 3: Some extended-type hypergeometric functions of two and three variables - p. 31 5. Chapter 4: An extension of the k-gamma and k-beta matrix functions by use of the two-parameter k-Mittag-Leffler function - p. 45 6. Chapter 5: A k-type Caputo fractional derivative operator and its properties - p. 63 7. Chapter 6: An extension of k-hypergeometric functions - p. 79 8. Chapter 7: Certain pathway fractional integral formulae involving extended k-hypergeometric functions - p. 107 9. Chapter 8: Certain integral transforms for the k-hypergeometric functions - p. 123 10. Chapter 9: The generalized Riemann–Liouville k-fractional derivative and its properties - p. 135 11. Chapter 10: Main results on U-Bernoulli–Korobov-type polynomials and their approximate roots - p. 149 12. Chapter 11: Solvability of the Cauchy problem for a fractionally loaded equation with variable coefficients - p. 163 13. Chapter 12: Inverse problem for the loaded heat conductivity equation with variable coefficients - p. 173 14. Chapter 13: Generalized arbitrary order Mittag-Leffler-type function and Marichev–Saigo–Maeda fractional operators - p. 189 15. Chapter 14: Fractional integration and differentiation of the generalized arbitrary order Mittag-Leffler-type function - p. 211 16. Chapter 15: Generalized fractional order kinetic equations involving multi-index Mittag-Leffler functions of several variables - p. 227 17. Chapter 16: Certain integrals involving k-type R- and G-functions - p. 243 18. Chapter 17: Fractional integrals and solutions of fractional kinetic equations involving R- and G-functions - p. 277 19. Index - p. 303
Extended Hypergeometric Functions and Orthogonal Polynomials presents a comprehensive and accessible resource for researchers and graduate students interested in exploring the rich connections between extended hypergeometric functions, orthogonal polynomials, and multivariable polynomials. Integrating all three fields and their applications in Maple, Mathematica, and MATLAB, this book fosters interdisciplinary understanding and inspires new avenues of research in mathematics, engineering, physics, and computer science. It also provides a glimpse into future research directions in these areas, including potential applications in emerging fields of applied mathematics and interdisciplinary collaborations. Each chapter begins with an introduction, includes sections on theory, followed by sections on applications, and ends with exercises, problems, references and suggested readings.
1. Introduction - p. xvii 2. Chapter 1: The generalized k-hypergeometric function as the Meijer G-function - p. 1 3. Chapter 2: Matrix analog of a new class of hypergeometric functions - p. 13 4. Chapter 3: Some extended-type hypergeometric functions of two and three variables - p. 31 5. Chapter 4: An extension of the k-gamma and k-beta matrix functions by use of the two-parameter k-Mittag-Leffler function - p. 45 6. Chapter 5: A k-type Caputo fractional derivative operator and its properties - p. 63 7. Chapter 6: An extension of k-hypergeometric functions - p. 79 8. Chapter 7: Certain pathway fractional integral formulae involving extended k-hypergeometric functions - p. 107 9. Chapter 8: Certain integral transforms for the k-hypergeometric functions - p. 123 10. Chapter 9: The generalized Riemann–Liouville k-fractional derivative and its properties - p. 135 11. Chapter 10: Main results on U-Bernoulli–Korobov-type polynomials and their approximate roots - p. 149 12. Chapter 11: Solvability of the Cauchy problem for a fractionally loaded equation with variable coefficients - p. 163 13. Chapter 12: Inverse problem for the loaded heat conductivity equation with variable coefficients - p. 173 14. Chapter 13: Generalized arbitrary order Mittag-Leffler-type function and Marichev–Saigo–Maeda fractional operators - p. 189 15. Chapter 14: Fractional integration and differentiation of the generalized arbitrary order Mittag-Leffler-type function - p. 211 16. Chapter 15: Generalized fractional order kinetic equations involving multi-index Mittag-Leffler functions of several variables - p. 227 17. Chapter 16: Certain integrals involving k-type R- and G-functions - p. 243 18. Chapter 17: Fractional integrals and solutions of fractional kinetic equations involving R- and G-functions - p. 277 19. Index - p. 303
