The generalized Hahn polynomials

In this paper, we represent the generalized Hahn polynomials ϕ(a)n (x, y) by the Cauchy operator for deriving its identities: generating function, Mehler’s formula, Rogers formula (with some of its applications), Rogers-type formula, extended generating function, extended Mehler’s formula, extended Rogers formula and another extended identities. Also, the Rogers-type formula for the bivariate (generalized) (classical) Rogers-Szeg¨o polynomials will be given by two methods. Then we give the q-integral representation for the generalized Hahn polynomials, bivariate Rogers-Szeg¨o polynomials, and the generalized Rogers-Szegö polynomials.Publisher's Versio

Doubly connected pitchfork domination and it’s inverse in graphs

Let G be a finite, simple, undirected graph and without isolated vertices. A subset D of V is a pitchfork dominating set if every vertex v ∈ D dominates at least j and at most k vertices of V − D, for any j and k integers. A subset Dˉ¹ of V − D is an inverse pitchfork dominating set if Dˉ¹ is a dominating set. The domination number of G, denoted by γpf (G) is a minimum cardinality over all pitchfork dominating sets in G. The inverse domination number of G, denoted by γˉ¹ pf (G) is a minimum cardinality over all inverse pitchfork dominating sets in G. In this paper, a special modified pitchfork dominations called doubly connected pitchfork domination and it’s inverse are introduced when j = 1 and k = 2. Some properties and bounds are studied with respect to the order and the size of the graph. These modified dominations are applied and evaluated for several well known and complement graphs.Publisher's Versio