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...

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 ...

Dawson's Integral is uðyÞ expðÀy 2 Þ R y 0 expðz 2 Þdz. We show that by solving the differential equation du=dy þ 2yu ¼ 1 using the orthogonal rational Chebyshev functions of the second kind, SB 2n ðy; LÞ, which generates a pentadiagonal Petrov–Galerkin matrix, one can obtain an accuracy of roughly ð3=8ÞN digits where N is the number of terms in the spectral series. The SB series is not as effi...

Recently, Babson and Steingrimsson (see [2]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on n letters avoiding 1-3-2 (or containing 1-3-2 exactly once) and an arbitrary generalized pattern τ on k letters, or containing τ exactly onc...

We study the existence and shape preserving properties of a generalized Bernstein operator Bn fixing a strictly positive function f0, and a second function f1 such that f1/f0 is strictly increasing, within the framework of extended Chebyshev spaces Un. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator Bn : C[a, b] → Un with strictly incr...

We discuss polynomial interpolation and derive sufficient conditions for the uniform convergence of Chebyshev interpolants for different classes of functions. Rigorous results are illustrated with a number of examples which include solitons on an infinite line with algebraic, exponential and Gaussian decay rates. Suitable mappings of the real line to the interval [−1, 1] are considered for each...

The Lane-Emden equation has been used to model several phenomenas in theoretical physics, mathematical physics and astrophysics such as the theory of stellar structure. This study is an attempt to utilize the collocation method with the Rational Chebyshev of Second Kind function (RSC) to solve the Lane-Emden equation over the semi-infinit interval [0,+∞). According to well-known results and com...

BACKGROUND Magnetic particle imaging (MPI) is a new tomographic imaging technique capable of imaging magnetic tracer material at high temporal and spatial resolution. Image reconstruction requires solving a system of linear equations, which is characterized by a "system function" that establishes the relation between spatial tracer position and frequency response. This paper for the first time ...