In this paper, we introduce a new class of generalized extended Laguerre-based Apostol-type-Bernoulli, Apostol-type-Euler and Apostoltype-Genocchi polynomials. These Apostol type polynomials are used to connect Fubini-Hermite Bell-Hermite find representations. We derive some implicit summation formulae symmetric identities for these families special functions by applying the generating functions.

In this study, we introduce sine and cosine Bell-based Frobenius-type Eulerian polynomials, by presenting several relations applications, analyze certain properties. Our first step is to obtain diverse formulas that cover summation formulas, addition with earlier polynomials in the literature, differentiation rules. Finally, after determining few zero values of draw graphical representations th...

We study the number P(G) of non-equivalent ways of coloring a given graph G. We show some similarities and differences between this graph invariant and the well known chromatic polynomial. Relations with Stirling numbers of the second kind and with Bell numbers are also given. We then determine the value of this invariant for some classes of graphs. We finally study upper and lower bounds on P(...

In the first half we show an interesting relation between coherent states and the Bell states in the case of spin 1/2, which was suggested by Fivel. In the latter half we treat generalized coherent states and try to generalize this relation to get several generalized Bell states. Our method is based on a geometry and our task may give a hint to open a deep relation between a coherence and an en...

Let K be a eld of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials. Crossed products of gro...

The parameterized pattern matching problem is to check if there exists a renaming bijection on the alphabet with which a given pattern can be transformed into a substring of a given text. A parameterized border array (p-border array) is a parameterized version of a standard border array, and we can efficiently solve the parameterized pattern matching problem using p-border arrays. In this paper...

A parametric down conversion source pumped by a monochromatic laser will produce frequency entangled photon pairs. We demonstrate this by an experiment in which five-dimensional frequency entanglement is manipulated at telecommunication wavelengths using commercially available components such as electro-optic phase modulators and narrowband frequency filters. A theoretical intuition for this ap...

A new family of generalized Stirling and Bell numbers is introduced by considering powers (V U)n of the noncommuting variables U, V satisfying UV = V U +hV s. The case s = 0 (and h = 1) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they a...

Keywords: Bell polynomial Divided difference Faà di Bruno's formula Multicomposite function a b s t r a c t The well-known formula of Faà di Bruno's for higher derivatives of a composite function has played an important role in combinatorics. In this paper we generalize the divided difference form of Faà di Bruno's formula and give an explicit formula for the n-th divided difference of a multic...

We describe a concrete construction of all pushout complements for two given morphisms f : A → B, m : B → D in the category of hypergraphs, valid also for the case where f, m are non-injective. To our knowledge such a construction has not been discussed before in the literature. It is based on the generation of suitable equivalence relations. We also give a combinatorial interpretation and show...