Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers

The sums of powers of arithmetic progressions is of the form a^p+(a+d)^p +(a+2d)^p+\cdots+(a+(n-1)d)^p, where n\geq 1, p is a non-negative integer, and a and d are complex numbers with d\neq 0. This sum can be computed using classical Eulerian numbers \cite{worpitzky1883studien} and general Eulerian numbers \cite{xiong2013general}. This paper gives a new method using classical Eulerian numbers to compute this sum. The existing formula that uses general Eulerian numbers are more algorithmically complex due to more numbers to compute. This paper presents and focuses on two novel algorithms involving both types of Eulerian numbers. This paper gives a comparison to Xiong \textit{et al.}’s result with general Eulerian numbers \cite{xiong2013general}. Moreover, an analysis of theoretical time complexities is presented to show our algorithm is less complex. Various values of p are analyzed in the proposed algorithms to add significance to the results. The experimental results show the proposed algorithm remains around 70\% faster as p increases.

Logistic Wavelets and Their Application to Model the Spread of COVID-19 Pandemic

In the present paper, we model the cumulative number of persons, reported to be infected with COVID-19 virus, by a sum of several logistic functions (the so-called multilogistic function). We introduce logistic wavelets and describe their properties in terms of Eulerian numbers. Moreover, we implement the logistic wavelets into Matlab’s Wavelet Toolbox and then we use the continuous wavelet transform (CWT) to estimate the parameters of the approximating multilogistic function. Using the examples of several countries, we show that this method is effective as a method of fitting a curve to existing data. However, it also has a predictive value, and, in particular, allows for an early assessment of the size of the emerging new wave of the epidemic, thus it can be used as an early warning method.

On the Joint Distribution of Descents and Signs of Permutations

We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.

An Order on Circular Permutations

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

Sign Variation and Descents

For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let $\Delta_{n,m}$ be the order complex of $P_{n,m}$. A previous result from the third author shows that $\Delta_{n,m}$ is Cohen-Macaulay over $\mathbb{Q}$ whenever $m$ is even or $m = n-1$. Hence, it follows that the $h$-vector of $\Delta_{n,m}$ consists of nonnegative entries. Our main result states that $\Delta_{n,m}$ is partitionable and we give an interpretation of the $h$-vector when $m$ is even or $m = n-1$. When $m = n-1$ the entries of the $h$-vector turn out to be the new Eulerian numbers of type $D$ studied by Borowiec and Młotkowski in [ Electron. J. Combin., 23(1):#P1.38, 2016]. We then combine our main result with Klee's generalized Dehn-Sommerville relations to give a geometric proof of some facts about these Eulerian numbers of type $D$.

On Fully Degenerate Daehee Numbers and Polynomials of the Second Kind

In a study, Carlitz introduced the degenerate exponential function and applied that function to Bernoulli and Eulerian numbers and degenerate special functions have been studied by many researchers. In this paper, we define the fully degenerate Daehee polynomials of the second kind which are different from other degenerate Daehee polynomials and derive some new and interesting identities and properties of those polynomials.