Each entry of these previously known determinant formulas is given as a finite linear combination of elementary/ complete symmetric

polynomials, while in our formula it is given as a possibly infinite linear combination of Grothendieck

polynomials associated to one row partitions.

Gautschi collects exercises and solutions from his textbook on orthogonal

polynomials in MATLAB, edits them slightly to make them self-contained, and adds many new ones.

A generalisation of synthetic division and a general theorem of division of

polynomials.

A class of

polynomials f(x) suiting our demands is introduced, and we showed their existence by numerical experiments.

The problem of stability of systems can be supposed as the problem of stability of their characteristic

polynomials.

The problem of division was reduced by using zeros of

polynomials to find inverse of a number modulo prime powers.

Tutte in 1954 in [22] as a generalization of chromatic

polynomials studied by Birkhoff [1] and Whitney [25].

A method to determine in an efficient way the Laurent

polynomials of Hermite interpolation is presented in [1].

Fomin and Green gave a version for certain non-commutative symmetric functions, which led to combinatorial formulas for characters of representations associated to stable Schubert and stable Grothendieck

polynomials [12].

Keywords: legendre associated functions, Legendre

polynomials, recurrence relations, stability.

n and each coefficient's commitment of added sum of

polynomials of f(x) and k(x) as follows: ([s.

Let C [x, y, z] be the ring of

polynomials in the variables x, y, z with coefficients in C.

Gegenbauer

polynomials or ultraspherical

polynomials [C.

This can be done for the unispherical windows [3] based on Gegenbauer

polynomials as well as for windows proposed by Zierhofer [4].

The more classical examples of linear positive operators throughout approximation process are the Bernstein

polynomials, which are defined by Bernstein [3] as following: