Here, in this article, we investigate the solution of a general family fractional-order differential equations by using spectral Tau method sense Liouville–Caputo type fractional derivatives with linear functional argument. We use Chebyshev polynomials second kind to develop recurrence relation subjected certain initial condition. The behavior approximate series solutions are tabulated and plot...

In [3] it has been shown that powers of the generating function c(x) of Catalan numbers {QaeNo ft ^ > > > 4 2 , •••}, w h e r e o : = {°, I> •••} (1 4 5 9 a n d A000108 of [8] and references of [3]) can be expressed in terms of a linear combination of 1 and c(x) with coefficients replaced by certain scaled Chebyshev polynomials of the second kind. In this paper, derivatives of c(x) are studied ...

In this work we analyze the boundary value problems on a path associated with Schrödinger operators with constant ground state. These problems include the cases in which the boundary has two, one or none vertices. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. Moreover, we obtain the Green’s function for each regular problem and th...

This paper contains two fast algorithms for inversion of ChebyshevvVander-monde matrices of the rst and second kind. They are based on special representations of the Bezoutians of Chebyshev polynomials of both kinds. The paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster, but also more stable than other algorithms avail...

The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax = b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for res...

Let fr n (k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let Fr(x; k) and F (x, y; k) be the generating functions defined by Fr(x; k) = ∑ n>0 f r n (k)xn and F (x, y; k) = ∑ r>0 Fr(x; k)y r . We find an explcit expression for F (x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 6 r 6 k via Che...

Let fr n(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let Fr(x; k) and F (x, y; k) be the generating functions defined by Fr(x; k) = P n>0 f r n(k)x n and F (x, y; k) = P r>0 Fr(x; k)y r. We find an explicit expression for F (x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 6 r 6 k via Cheb...

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the e...

We study the total positivity of the kernel 1/(x+2 cos(πα)xy+y). The case of infinite order is characterized by an application of Schoenberg’s theorem. We then give necessary conditions for the cases of any given finite order with the help of Chebyshev polynomials of the second kind. Sufficient conditions for the finite order cases are also obtained, thanks to Propp’s formula for the Izergin-Ko...

We study generating functions for the number of involutions in Sn avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation τ on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and 12 . . . k have the ...