On the Watson L₂-theory for index transforms II

Abstract

We continue to investigate index transforms which are associated with general Kontorovich–Lebedev kernels (GKL kernels). The case of the so-called extended Watson kernels is considered. An inversion theorem is proved. Examples of integral transformations by the index of Whittaker's functions and Meijer's G-functions are demonstrated. As applications, a general index integral involving a product of Meijer's G-functions is calculated.

Keywords:

• index transforms
• Kontorovich–Lebedev transform
• Watson transforms
• Watson kernels
• modified Bessel functions
• Meijer's G-function
• Whittaker's functions
• Parseval equality
• Plancherel theorem

2000 Mathematics Subject Classification :

• Primary: 44A15
• 33C10
• 33C40

Acknowledgement

The present investigation was supported, in part, by the ‘Centro de Matemática’ of the University of Porto.