The generalized Stirling numbers Ss;h(n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k;α, β, r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n, k) and the associated Bell numbers Bs;h(n) and Bell polynomials Bs;h|n(x) are derived. The particular case s = 2 and h = −1 corres...

(About Jackson q-Bessel functions) Laplace transform allows to resolve differential equations in the neighborhood of an irregular singular point. The purpose of the article is to study how to apply a basic Borel– Laplace transformation to q-difference equations satisfied by the q-Bessel functions of F.H. Jackson. Connection matrices are obtained between solutions at the origin and solutions at ...

Bessel beams have been extensive studied to date but are always created over a finite region inside the laboratory. Means to overcome this consider multi-element refractive designs to create beams that have a longitudinal dependent cone angle, thereby allowing for a far greater quasi non-diffracting propagation region. Here we outline a generalized approach for the creation of shape-invariant B...