A generalized Milne-Thomson theorem

Using analytic continuation theory, a new simple proof of a standard generalized circle theorem is given. Additionally, new cases involving complex coefficients in the boundary condition and allowing for an arbitrary singularity of a given complex potential at the interface are considered. © 2005 Elsevier Ltd. All rights reserved

A generalized Milne-Thomson theorem for the case of parabolic inclusion

Complex analysis methods are applied to determine a velocity field of seepage in a heterogeneous infinite planar medium consisting of two dissimilar homogeneous components with a parabolic interface. New cases with arbitrary singularities of the principal part of a required complex potential are considered. © 2008 Elsevier Inc. All rights reserved

Bernoulli type polynomials on Umbral Algebra

The aim of this paper is to investigate generating functions for modification of the Milne-Thomson's polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. By applying the Umbral algebra to these generating functions, we provide to deriving identities for these polynomials

Vortex Images and q-Elementary Functions

In the present paper problem of vortex images in annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, where dimensionless parameter $q = r^2_2/r^2_1$ is given by square ratio of the cylinder radii. Resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we link our solution with result of Johnson and McDonald. We found that one vortex cannot remain at rest except at the geometric mean distance, but must orbit the cylinders with constant angular velocity related to q-harmonic series. Vortex images in two particular geometries in the $q \to \infty$ limit are studied.Comment: 17 page

Biorthogonality of the Lagrange interpolants

We show that the Lagrange interpolation polynomials are biorthogonal with respect to a set of rational functions whose poles coinicde with interpolation point

On certain generalized q-Appell polynomial expansions

We study q-analogues of three Appell polynomials, the H-polynomials, the Apostol–Bernoulli and Apostol–Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials