This paper presents novel analytic expressions for the Rice Ie−function, Ie(k, x), and the incomplete Lipschitz-Hankel Integrals (ILHIs) of the modified Bessel function of the first kind, Iem,n(a, z). Firstly, an exact infinite series and an accurate polynomial approximation are derived for the Ie(k, x) function which are valid for all values of k. Secondly, an exact closed-form expression is d...

—Pointwise-supported generalised wavelets are introduced, based on Dirac, doublet and further derivatives of (t). A generalised biorthogonal analysis leads to standard Taylor series and new Dual-Taylor series that may be interpreted as Laurent Schwartz distributions. A Parseval-like identity is also derived for Taylor series, showing that Taylor series support an energy theorem. New representa...

In this study integral form and new recursive formulas for Favard constants and some connected with them numeric and Fourier series are obtained. The method is based on preliminary integration of Fourier series which allows for establishing finite recursive representations for the summation. It is shown that the derived recursive representations are numerically more effective than known represe...

Ordinal time series analysis is a new approach to the qualitative investigation of long and complex time series. The idea behind it is to transform a given time series into a series of ordinal patterns each describing the order relations between the present and a fixed number of equidistant past values at a given time. Here we consider ordinal pattern distributions and some measures derived fro...

P' takes the place of P, and the new elements are functions of the original elements, /JL is determined from the condition that the new series must be of the same general form as the old. If, in addition, x be replaced by 1/x another series is obtained. From these two new series, by proper substitution of the new derived elements, are obtained, almost by inspection, the twentyfour different ser...

which holds for mi non-negative integers and Re (a + |m|) < 1. Accordingly, hypergeometric series with integral parameter differences have been called Karlsson–Minton type hypergeometric series; other results for such series may be found in [C, G1, G2, S1, S2]. In recent work [R], we have derived a very general reduction formula for series of Karlsson–Minton type. We recall it in its most gener...

Statistical differentiability of the measure along the reconstructed trajectory is a good candidate to quantify determinism in time series. The procedure is based upon a formula that explicitly shows the sensitivity of the measure to stochasticity. Numerical results for partially surrogated time series and series derived from several stochastic models, illustrate the usefulness of the method pr...

The finite-lattice method of series expansions has been combined with a new extrapolation procedure to extend the low-temperature series for the specific heat, spontaneous magnetisation and susceptibility of the spin-1 Ising model on the square lattice. The extended series were derived by directly calculating the series to order 99 (in the variable u = exp[−J/k B T ]) and using the new extrapol...