A probabilistic interpretation of a modified Gegenbauer polynomial is furnished by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinat...

An operational approach introduced by Gould and Hopper to the construction of generalized Hermite polynomials is followed in the hypercomplex context to build multidimensional generalized Hermite polynomials by the consideration of an appropriate basic set of monogenic polynomials. Directly related functions, like Chebyshev polynomials of first and second kind are constructed.

In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix of fractional derivative of order $ gamma $ in the Caputo for FCSs and show that this matrix with the Tau method are utilized to reduce the solution of some fractional-order differential equations.

In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we transform the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equ...

with given a, b, t0, t1 and n ≥ 0. This sequence was introduced by Horadam [3] in 1965, and it generalizes many sequences (see [1, 4]). Examples of such sequences are Fibonacci polynomials sequence (Fn(x))n≥0, Lucas polynomials sequence (Ln(x))n≥0, and Pell polynomials sequence (Pn(x))n≥0, when one has a = x, b = t1 = 1, t0 = 0; a = t1 = x, b = 1, t0 = 2; and a = 2x, b = t1 = 1, t0 = 0; respect...

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that Tn(c cos θ) and Un(c cos θ) are positive definite functions. We further s...

in this study, the problem of determining an optimal trajectory of a nonlinear injection into orbit problem with minimum time was investigated. the method was based on orthogonal polynomial approximation. this method consists of reducing the optimal control problem to a system of algebraic equations by expanding the state and control vector as chebyshev or legendre polynomials with undetermined...

In paper [4], transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is rem...

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to mixed conditions on interval 0≤x≤1 was considered. First, BVP converted into an equivalent differential–integral merging conditions. Then, shifted Chebyshev polynomials and collocation method were used solve equation. Varying coefficients also decomposed truncated series s...