In this paper, we investigate the fractional power $(-{\mathrm{\Delta }}_{\nu }{)}^{\gamma /2}$, $0<\gamma <2,$ of the Bessel operator in the form ${\mathrm{\Delta }}_{\nu }:=\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\frac{2\nu +1}{x}\frac{\mathrm{d}}{\mathrm{d}x},\nu >-1/2.$ Our method uses a canonical representation for generalized Laplacian related to the structures of the Bessel–Kingman hypergroup. As a direct application, we solve the space-fractional diffusion equation.

The aim of this work is to establish in great detail The q-Fourier analysis related to the q-cosine. The wise reader will note that the considered q-cosine coincides with the one given by T.H. Koornwinder and S.F. Swarttouw. Through the q-cosine product formula, we define and analyze the properties of the q-even translation and the q-convolution. Adopting the Titchmarsh approach, we study the q-cosine Fourier transform and its inverse formula. The second theme of this paper is an application of the q-Fourier analysis developed earlier. We extend the heat representation theory inaugurated by P.C. Rosenbloom and D.V. Widder to the q-analogue. We construct the q-solution source, the q-heat polynomials and solve the q-analytic Cauchy problem.

In this paper, we apply a limit as n→∞ from the little−1 Jacobi polynomials operator to obtain a new singular first-order differential-difference operator Y_α on the real line. The eigenfunction of this operator extends the Hartley kernel. Also, we give a special case of the Jordan algebra that has the operator Y_α as one of their three generators. Further, we study some elements of harmonic analysis related to Y_α.

We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J₁-topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.