Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice

We complete the construction of raising and lowering operators, given in a previous work, for the orthogonal polynomials of hypergeometric type on non-homogeneous lattice, and extend these operators to the generalized orthogonal polynomials, namely, those difference of orthogonal polynomials that satisfy a similar difference equation of hypergeometric type.Comment: LaTeX, 19 pages, (late submission to arXiv.org

On the Properties of Special Functions on the linear-type lattices

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference-recurrence relations for such functions. Finally, applications to $q$-classical polynomials are given

On the limit of non-standard q-Racah polynomials

The aim of this article is to study the limit transitions from non-standard q-Racah polynomials to big q-Jacobi, dual q-Hahn, and q-Hahn polynomials such that the orthogonality properties and the three-term recurrence relations remain valid

On a problem of Erd\H{o}s and Rothschild on edges in triangles

Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best possible in terms of the range for C, as it is known that every N-vertex graph with more than (N^2)/4 edges contains an edge that is in at least N/6 triangles.Comment: 8 page

Influence of air diffusion on the OH radicals and atomic O distribution in an atmospheric Ar (bio)plasma jet

Treatment of samples with plasmas in biomedical applications often occurs in ambient air. Admixing air into the discharge region may severely affect the formation and destruction of the generated oxidative species. Little is known about the effects of air diffusion on the spatial distribution of OH radicals and O atoms in the afterglow of atmospheric-pressure plasma jets. In our work, these effects are investigated by performing and comparing measurements in ambient air with measurements in a controlled argon atmosphere without the admixture of air, for an argon plasma jet. The spatial distribution of OH is detected by means of laser-induced fluorescence diagnostics (LIF), whereas two-photon laser-induced fluorescence (TALIF) is used for the detection of atomic O. The spatially resolved OH LIF and O TALIF show that, due to the air admixture effects, the reactive species are only concentrated in the vicinity of the central streamline of the afterglow of the jet, with a characteristic discharge diameter of similar to 1.5 mm. It is shown that air diffusion has a key role in the recombination loss mechanisms of OH radicals and atomic O especially in the far afterglow region, starting up to similar to 4mm from the nozzle outlet at a low water/oxygen concentration. Furthermore, air diffusion enhances OH and O production in the core of the plasma. The higher density of active species in the discharge in ambient air is likely due to a higher electron density and a more effective electron impact dissociation of H2O and O-2 caused by the increasing electrical field, when the discharge is operated in ambient air

Mathematical Structure of Relativistic Coulomb Integrals

We show that the diagonal matrix elements $,$ where $O$ $={1,\beta,i\mathbf{\alpha n}\beta}$ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, may be considered as difference analogs of the radial wave functions. Such structure provides an independent way of obtaining closed forms of these matrix elements by elementary methods of the theory of difference equations without explicit evaluation of the integrals. Three-term recurrence relations for each of these expectation values are derived as a by-product. Transformation formulas for the corresponding generalized hypergeometric series are discussed.Comment: 13 pages, no figure

Characterization of a planar 8 mm wide radiofrequency atmospheric pressure plasma source by spectroscopy techniques

Atmospheric pressure planar radiofrequency (RF) 13.56 MHz discharge in Ar gas generated in a long gap is investigated. The discharge operation with and without a dielectric barrier on the electrodes is studied as a function of the applied power and gas flow. The source afterglow is characterized and is analyzed for possible large-scale biomedical applications where low gas temperature is required. The discharge is studied by relative and absolute emission spectroscopies. A gas temperature as low as 330 +/- 50 K is determined from the rotational-vibrational band of OH emission. The absolute value of the discharge continuum irradiation is used to determine the electron density and the electron temperature. The electron-atom and electron-ion contributions to the bremsstrahlung radiation are calculated and are compared with measured spectra. The electron density of 1.9 +/- 1 x 10(20) m(-3) and electron temperature of 1.75 +/- 0.25 eV are measured in the discharge without a dielectric barrier. It is found that presence of the dielectric has a negligible effect on electron temperature, whereas the electron number density is almost six times lower in the discharge with the dielectric barrier