Harmonic series with polylogarithmic functions

Introduction/purpose: Some sums of the polylogarithmic function associated with harmonic numbers are established. Methods: The approach is based on using the summation methods. Results: This paper generalizes the results of the zeta function series associated with the harmonic numbers. Conclusions: Various interesting series as the consequence of the generalization are obtained.

ON A FAMILY OF (p, q)-HYBRID POLYNOMIALS

In this paper, the class of (p,q)-Bessel-Appell polynomials is introduced. The generating function, series definition and determinant definition of this class are established. Certain members of (p,q)-Bessel-Appell polynomials are considered and some properties of these members are also derived. Further, the class of 2D (p,q)-Bessel-Appell polynomials is introduced by means of the generating function and series definition. In addition, the graphical representations of some members of (p,q)-Bessel-Appell polynomials and 2D (p,q)-Bessel-Appell polynomials are plotted with the help of Matlab.

k Part Partitions of Integers

This paper presents formulas (together with their proofs) determining 3 and 4 part partitions of any integer. These formulas were derived using the properties of the floor function and Bernoulli formulas for various powers of finite sums of the floor function series. This made it possible to obtain above formulas in a much simpler way than most traditional methods.

Orthogonal Functions for Evaluating Social Distancing Impact on CoVID-19 Spread

Early CoVID-19 growth often obeys: N{t}=N/I\exp[+K/o\t], with K/o\=[(ln2)/(t/dbl\)], where t/dbl\ is the pandemic doubling time, prior to society-wide Social Distancing. Previously, we modeled Social Distancing with t/dbl\ as a linear function of time, where N[t]=1exp[+K/A\ t/(1+ gamma/o\ t)] is used here. Additional parameters besides {K/o\,gamma/o\} are needed to better model different rho[t]=dN[t]/dt shapes. Thus, a new Orthogonal Function Model [OFM] is developed here using these orthogonal function series: N(Z) = sum[m=0,M/F\] g/m\ L/m\(Z) exp[-Z] , R(Z) = sum[m=0,M/F\] c/m\ L/m\(Z) exp[-Z] , where N(Z) and Z[t] form an implicit N[t]=N(Z[t]) function, giving: G/o\ = [K/A\ / gamma/o\ ] , Z[t] = +[ G/o\ / (1+ gamma/o\t) ] , rho[t] = [ gamma/o\ / G/o\ ] (Z^2) R(Z) , with L/m\(Z) being the Laguerre Polynomials. At large M/F\ values, nearly arbitrary functions for N[t] and rho[t]=dN[t]/dt can be accommodated. How to determine {K/A\, gamma/o\} and the {g/m\; m=(0,+M/F\)} constants from any given N(Z) dataset is derived, with rho[t] set by: c/(M/F\ - k)\ = sum[m=0,k] g/m\ . The bing.com USA CoVID-19 data was analyzed using M/F\=(0,1,2) in the OFM. All results agreed to within about 10 percent, showing model robustness. Averaging over all these predictions gives the following overall estimates for the number of USA CoVID-19 cases at the pandemic end: <N/max\> = 5,009,677 (+/-) 269,450 (data to 5/3/20), and <N/max\> = 4,422,803 (+/-) 162,580 (data to 6/7/20), which compares the pre- and post-early May bing.com revisions. The CoVID-19 pandemic in Italy was examined next. The M/F\=2 limit was inadequate to model the Italy rho[t] pandemic tail. Thus, regions with a quick CoVID-19 pandemic shutoff may have additional Social Distancing factors operating, beyond what can be easily modeled by just progressively lengthening pandemic doubling times (with 13 Figures).

Symmetries in Hidden Bifurcation Routes to Multiscroll Chaotic Attractors Generated by Saturated Function Series

In this paper, hidden bifurcation routes to multiscroll chaotic attractors generated by saturated function series are explored. The method to nd such hidden bifurcation routes (HBR) depending upon two parameters is similar to the method introduced by Menacer, et al. (2016) for Chua multiscroll attractors. These HBR are characterized by the maximal range extension (MARE) of their attractors and coding the appearance order of the scrolls under the control of the two parameters. Moreover, these HDR have interesting symmetries with respect to the two parameters. The novelty that this article introduces, is firstly the paradigm of MARE and the formula giving their approximate value depending upon parameters p and q, which is linked to the size of the scrolls; secondly the coding of the HBR which is dened for the first time including the basic cell; and thirdly unearthing the symmetries of these routes, allowing to obtain their coding without any numerical computation.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Certain Results for the Twice-Iterated 2D q-Appell Polynomials

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated.