The vector-valued big q-Jacobi transform

Big $q$-Jacobi functions are eigenfunctions of a second order $q$-difference operator $L$. We study $L$ as an unbounded self-adjoint operator on an $L^2$-space of functions on $\mathbb R$ with a discrete measure. We describe explicitly the spectral decomposition of $L$ using an integral transform $\mathcal F$ with two different big $q$-Jacobi functions as a kernel, and we construct the inverse of $\mathcal F$.Comment: 35 pages, corrected an error and typo

Anomalous diffusion and the first passage time problem

We study the distribution of first passage time (FPT) in Levy type of anomalous diffusion. Using recently formulated fractional Fokker-Planck equation we obtain three results. (1) We derive an explicit expression for the FPT distribution in terms of Fox or H-functions when the diffusion has zero drift. (2) For the nonzero drift case we obtain an analytical expression for the Laplace transform of the FPT distribution. (3) We express the FPT distribution in terms of a power series for the case of two absorbing barriers. The known results for ordinary diffusion (Brownian motion) are obtained as special cases of our more general results.Comment: 25 pages, 4 figure

On a quasi-periodic Hopf bifurcation

In this paper we study quasi-periodic Hopf bifurcations for the model problem of a quasi-periodically forced oscillator, where the frequencies remain fixed. For this purpose we first consider Stoker's problem for small damping

Singular Linear Differential Equations in Two Variables

The formal and analytic classification of integrable singular linear differential equations has been studied. We provide a simple proof of the main result, namely: For certain irregular systems in two variables there is no Stokes phenomenon, i.e. there is no difference between the formal and the analytic classification.