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.
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