The Hypergeometric Approach to Integral Transforms and Convolutions

1. Front Matter

   Pages i-xi

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2. Preliminaries

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 1-14

3. Mellin Convolution Type Transforms With Arbitrary Kernels

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 15-40

4. H- and G-transforms

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 41-58

5. The Generalized H- and G-transforms

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 59-68

6. The Generating Operators of Generalized H-transforms

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 69-78

7. The Kontorovich-Lebedev Transform

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 79-108

8. General W-transform and its Particular Cases

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 109-138

9. Composition Theorems of Plancherel Type for Index Transforms

   • Semen B. Yakubovich, Yurii F. Luchko

   Pages 139-148

10. Some Examples of Index Transforms and Their New Properties

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 149-166

11. Applications to Evaluation of Index Integrals

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 167-172

12. Convolutions of Generalized H-transforms

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 173-182

13. Generalization of the Notion of Convolution

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 183-188

14. Leibniz Rules and Their Integral Analogues

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 189-204

15. Convolutions of Generating Operators

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 205-212

16. Convolution of the Kontorovich-Lebedev Transform

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 213-228

17. Convolutions of the General Index Transforms

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 229-240

18. Applications of the Kontorovich-Lebedev type Convolutions to Integral Equations

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 241-252

19. Convolutional Ring C_α

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 253-264

20. The Fields of the Convolution Quotients

    • Semen B. Yakubovich, Yurii F. Luchko

    Pages 265-276

The aim of this book is to develop a new approach which we called the hyper­ geometric one to the theory of various integral transforms, convolutions, and their applications to solutions of integro-differential equations, operational calculus, and evaluation of integrals. We hope that this simple approach, which will be explained below, allows students, post graduates in mathematics, physicists and technicians, and serious mathematicians and researchers to find in this book new interesting results in the theory of integral transforms, special functions, and convolutions. The idea of this approach can be found in various papers of many authors, but systematic discussion and development is realized in this book for the first time. Let us explain briefly the basic points of this approach. As it is known, in the theory of special functions and its applications, the hypergeometric functions play the main role. Besides known elementary functions, this class includes the Gauss's, Bessel's, Kummer's, functions et c. In general case, the hypergeometric functions are defined as a linear combinations of the Mellin-Barnes integrals. These ques­ tions are extensively discussed in Chapter 1. Moreover, the Mellin-Barnes type integrals can be understood as an inversion Mellin transform from the quotient of products of Euler's gamma-functions. Thus we are led to the general construc­ tions like the Meijer's G-function and the Fox's H-function.

• Cauchy problem
• Integral equation
• convolution
• differential equation
• differential operator
• integral transform
• operational calculus

• Department of Mathematics and Mechanics, Beylorussian State University, Minsk, Byelorussia, Russia

  Semen B. Yakubovich, Yurii F. Luchko

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