We introduce the poly-Cauchy polynomials which generalize the classical Cauchy polynomials, and investigate their arithmetical and combinatorial properties. These polynomials are considered as analogues of the poly-Bernoulli polynomials that generalize the classical Bernoulli polynomials. Moreover, we investigate the zeta functions which interpolate the poly-Cauchy polynomials. The values of th...

Let μ be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, · · · , x, . . . } to get a sequence {Pn}∞n=0 of orthogonal polynomials with respect to μ. In this paper we explain a method of deriving a generating function ψ(t, x) for μ. The power series expansion of ψ(t, x) in t produces the explicit form of p...

The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of this algorithms is based on the approximate calculation of higher derivatives using well-known finite-difference technique for the partial differential equations. The approximate solution is giv...

We describe an algebro-geometric construction of polygon-bounded minimal surfaces in ADS5, based on consideration of what we call the ”boundary ring” of polynomials. The first non-trivial example of the Nambu-Goto (NG) solutions for Z6-symmetric hexagon is considered in some detail. Solutions are represented as power series, of which only the first terms are evaluated. The NG equations leave a ...

The relationship between truncated Fourier series solutions of several different long-wave evolution equations is explored. The models are all designed to represent the same physics in the asymptote of small nonlinearity and frequency dispersion, and yet give different numerical solutions for given values of dimensionless parameters. It is found that the problem of obtaining a steady solution i...

Earlier work has shown how to recognize handwritten characters by representing coordinate functions or integral invariants as truncated orthogonal series. The series basis functions are orthogonal polynomials defined by a Legendre-Sobolev inner product. It has been shown that the free parameter in the inner product, the “jet scale”, has an impact on recognition both using coordinate functions a...

In this work, we will compare two approximation method based on hybrid Legendre andBlock-Pulse functions and a computational method for solving nonlinear Fredholm-Volterraintegral equations of the second kind which is based on replacement of the unknown functionby truncated series of well known Block-Pulse functions (BPfs) expansion