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In this paper, we use particular polynomials to establish some results on the real rootedness of polynomials. The considered polynomials are Bell polynomials and Hermite polynomials.

AbstractIn this paper, we use differential operator to present new identities and provide alternative proofs for certain established identities related to geometric polynomials. Additionally, by using the differential operator associated with geometric polynomials, Rolle’s theorem and a theorem of Wang and Yeh, we present some results on the real rootedness of polynomials. [ABSTRACT FROM AUTHOR]

Summary: Despite the close of a tumultuous 2020 and the start of 2021, connected devices will continue to shape the future of numerous industries, and businesses are confident that the Internet of Things (IoT) will play a key role in the future success of their trade. The growing Internet of Things (IoT) is connecting devices to a variety of sensors, applications, and other IoT elements to automate business processes and support human efficiencies in business and the home. WSN along with node localization algorithms can play a critical role in IoT applications. Nevertheless, in IoT applications, the context of real‐time location‐based services is gaining an overwhelming interest. To do this, several approaches are proposed in the recent literature based mainly on computational intelligence algorithms. This paper proposes a node localization algorithm based on swarm intelligence algorithms, that is, a hybrid Harris Hawks optimization based on differential evolution (HHODE).HHODE algorithm relies on Euclidian Distance as objective function to evaluate best‐fit coordinates of sensor nodes in a wireless sensor network. Moreover, several experimentations are performed depending on the network size, communication range of sensors, geographical distribution, and the beacon nodes' density to demonstrate the efficiency of the HHODE algorithm. Compared to Standard DE, HOO, PSO, and Bat Algorithm, HHODE shows higher performance with regard to node localization. [ABSTRACT FROM AUTHOR]

Let p be a prime greater than or equal to 5: In this paper, by using the harmonic numbers and Fermat quotient we establish congruences involving the sums... [ABSTRACT FROM AUTHOR]

In this paper, by the differential operator we give some congruences and new proof for some known congruence concerning the Bell polynomials, derangement polynomials and r-Lah polynomials. [ABSTRACT FROM AUTHOR]

In this paper, by the classical umbral calculus method, we establish identities involving the Appell polynomials and extend some existing identities. [ABSTRACT FROM AUTHOR]

Congruences involving sums of Harmonic numbers and binomial coefficients are considered in this paper. Recently, many great mathematicians have been interested to find congruences and relationships between these numbers such Sun & Tauraso, Koparal & Ömür, Mao & Sun and Meštrović & Andjić. In the present paper, some new combinatorial congruences are proved. These congruences are mainly determined modulo p² or p³ (in any prime) and they are motivated by a recent paper by Meštrović and Andjić. The first main result (Theorem 1) presents the congruence modulo p² (p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. Two interesting congruences modulo a prime p > 3 (Corollary 2) involving Harmonic numbers Hk, Catalan numbers ck and Fermat quotient q2 := (2p-1 - 1)/p are obtained as consequences of Theorem 1. The second main result (Theorem 2) presents the congruence modulo p³(p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. [ABSTRACT FROM AUTHOR]

The main object of this paper is to express the values at non-negative integers of the generalized Bernoulli polynomials by using a class of the Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of sequences of polynomials linked to the sequence of Bell polynomials. Some sequences of this class have applications on the theory of hyperbolic differential equations and other sequences generalize Laguerre polynomials and associated Lah polynomials. We discuss, for these polynomials, their explicit expressions, relations to the successive derivatives of a given function, real zeros and recurrence relations. Some known results are significantly simplified. [ABSTRACT FROM AUTHOR]

Let G be a threshold graph. In this paper, we give, in first hand, a formula relating the chromatic polynomial of G (the complement of G) to the chromatic polynomial of G: In second hand, we express the chromatic polynomials of G and G in terms of the generalized Bell polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the umbral representation of the geometric polynomials wxn; = wn (x) to derive some properties involving these polynomials. Furthermore, for any prime number p and any polynomial f with integer coefficients, we show (f (wx))p≡ f (wx) (mod p) and we give other curious congruences. [ABSTRACT FROM AUTHOR]

In this paper we give some congruences on the r-derangement polynomials (defined below), Lah polynomials and some versions of Bell numbers and polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the r-Jacobi-Stirling numbers of the second kind introduced by Gelineau in his Phd thesis. We give, upon using combinatorial and analytic arguments, the ordinary generating function of these numbers, two recurrence relations, their exact expressions and the log-concavity. [ABSTRACT FROM AUTHOR]

Let be subsets of the finite set with and for all , . The ( r1,...,rp)-Stirling number of the second kind, introduced in this paper and denoted by , counts the number of partitions of the set into kclasses (or blocks) such that the elements in each , , are in different classes (or blocks). Combinatorial and algebraic properties of these numbers are explored. [ABSTRACT FROM AUTHOR]

Chou, Hsu and Shiue gave some applications of Faà di Bruno's formula to characterize inverse relations. Our aim is to develop some inverse relations connected to the multipartitional type polynomials involving to binomial type sequences. [ABSTRACT FROM AUTHOR]

In a previous paper we investigated the (exponential) bipartitional polynomials involving polynomial sequences of trinomial type. Our aim is to give properties of bipartitional polynomials related to the derivatives of polynomial sequences of trinomial type. Furthermore, we deduce identities involving Bell polynomials. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate and present the general properties of the (exponential) bipartitional polynomials. After establishing relations between bipartitional polynomials and polynomial sequences of binomial and trinomial type, a number of identities are deduced. [ABSTRACT FROM AUTHOR]

A set of vertices S in a graph G = (V,E) is a dominating set if every vertex not in S is adjacent to at least one vertex in S. A domatic partition of graph G is a partition of its vertex-set V into dominating sets. A domatic partition P of G is called b-domatic if no larger domatic partition of G can be obtained from P by transferring some vertices of some classes of P to form a new class. The minimum cardinality of a b-domatic partition of G is called the b-domatic number and is denoted by bd(G). In this paper, we explain some properties of b-domatic partitions, and we determine the b-domatic number of some families of graphs. [ABSTRACT FROM AUTHOR]