Abstract
We consider the operator , where is the three term recurrence relation for the Jacobi polynomial normalized at the point x = 1 and I is the identity operator acting on . Knowing that for , these polynomials satisfy a product formula which provides a hypergroup structure on , we use the tools of harmonic analysis on this hypergroup to solve the space-time fractional diffusion equation Here, , is the Caputo time fractional derivative of order σ. The solution is given by , being the fundamental solution expressed in terms of Mittag-Leffler function and is the convolution on the hypergroup . We prove that, for , is nonnegative and we give some of its properties. Next, we study the one parameter operators associated with this fundamental solution.
Acknowledgments
The author is grateful to the referee for his valuable comments.
Disclosure statement
No potential conflict of interest was reported by the author(s).