The series $\sum_{k=1}^{\infty}\frac{k^{2^l}}{3^k}$ is the case z = $\frac{1}{3}$, $s = -2^l$ of the polylogarithm function $\mathrm{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$. We show that the value of the series is an integer if $l$ is an integer greater than one.

Recently the Legendre and other Sturm-Liouville (SL) polynomials were found as eigenvectors of certain matrices [2, 3, 4, 5]. However, the proposed algorithms are computationally incomplete and do not lead to general formulas to calculate the coefficients of SL polynomials of any order. In this paper, we complete the algorithms based on a matrix-eigenvector method, which can be used to compute SL polynomials of any order. This includes Legendre, Hermite, Laguerre, and Chebyshev polynomials.

In recent years, the Legendre and other Sturm-Liouville polynomials have been computed using matrix-eigenvalue algorithms. In the present work, we extend the Legendre matrix-eigenvalue algorithm to compute the associated Legendre polynomials, which have broad applications in a large variety of fields including quantum mechanics and computer science.

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid 1960s. Graph labeling is one of the fascinating areas of graph theory with wide ranging applications. A collection $\mathcal{A}$ of isomorphic copies of a given subgraph $G$ of $T$ is said to be an orthogonal double cover (ODC) of the graph $T$ by $G$, if every edge of $T$ belongs to exactly two members of $\mathcal{A}$ and any two different elements from $\mathcal{A}$ share at most one edge. The existence of an orthogonal labeling of a graph $G$ with respect to a certain group implies the existence of the cyclic orthogonal double cover of the circulant graphs on that group. In this paper, we prove the existence of orthogonal labeling for some different infinite graph classes and hence, the existence of the cyclic orthogonal double cover of some different infinite circulant graphs.

This paper studies the relationship between the major index and inversion number for Fibonacci compositions, that is compositions with parts 1 and 2 only. We prove the two statistics are equidistributed for this class of compositions.

In this paper, we define the notion of topological lattice effect algebras and investigate some of their properties. By using Sasaki arrows and F-balls, we construct two topology on lattice effect algebras. Then we study separation axioms on lattice effect algebras. Specifically, we find some conditions under which a topological lattice algebra is a $T_{0}$, $T_{1}$, and Hausdorff space. Finally, by using a strong filter and a quotient lattice effect algebra constructed by it, we investigate under what conditions this quotient lattice effect algebra will be a topological lattice effect algebra.

In this paper, we introduce the notion of weak AB-algebras, which are a generalization of BCC-algebras. The concepts of congruences and formation of quotients of these algebras are demonstrated. In addition, we prove the fundamental theorems of isomorphism for weak AB-algebras.

The concept of picture fuzzy sets was first considered by Cuong and Kreinovich [6] in 2013, which are direct extensions of fuzzy sets and intuitionistic fuzzy sets. In this paper, we applied the concept of picture fuzzy sets in UP-algebras to introduce the eight new concepts of picture fuzzy sets: picture fuzzy UP-subalgebras, picture fuzzy near UP-filters, picture fuzzy UP-filters, picture fuzzy implicative UP-filters, picture fuzzy comparative UP-filters, picture fuzzy shift UP-filters, picture fuzzy UP-ideals, and picture fuzzy strong UP-ideals. The link between the eight new concepts of picture fuzzy sets in UP-algebras is also discussed.

For $a \neq 1$ and $p$ a prime, we define numbers of the form $pa^2$ to be Square-Prime (SP) numbers. For example, $75 = 3 \cdot 25$; $108 = 3 \cdot 36$; $45 = 5 \cdot 9$. These numbers are listed in the OEIS as A228056. We study the properties of these numbers, their distribution/density, and also develop a few claims on their distribution/density. We rely on computer programs to verify some conjectures up to large numbers.