One expansion of the chromatic polynomial n(G,x) of a graph G relies on spanning trees of a graph. In fact, for a connected graph G of order n, one can express n(G,x) = (1 )“-‘x cyi’=;’ ti (1 -x)‘, where ti is the nun&r of spanning trees with external activity 0 and internal activity i. Moreover, it is known (via commutative ring theory) that ti arises as the number of monomials of degree n i 1...

We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with ...

In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion form...

Abstract.In this paper, the generalized Bessel matrix polynomials are introduced, starting from the hypergeometric matrix function. Integral form, Rodrigues’s formula and generating matrix function are then developed for the generalized Bessel matrix polynomials. These polynomials appear as finite series solutions of second-order matrix differential equations and orthogonality property for the ...

In this paper, we introduce a family of fractional-order Chebyshev functions based on the classical Chebyshev polynomials. We calculate and derive the operational matrix of derivative of fractional order $gamma$ in the Caputo sense using the fractional-order Chebyshev functions. This matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...

Complex Multiplication (CM) method is a frequently used method for the generation of prime order elliptic curves (ECs) over a prime field Fp. The most demanding and complex step of this method is the computation of the roots of a special type of class polynomials, called Hilbert polynomials. These polynonials are uniquely determined by the CM discriminant D. The disadvantage of these polynomial...

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with posit...

Optimal stability polynomials are polynomials whose stability region is as large as possible in a certain region, here the real-negative axis. We are interested in such polynomials wich in addition, obey a certain order condition. An important application of these polynomials is the construction of stabilized explicit Runge-Kutta methods (see 6] for a survey article and 4]). In this paper we wi...