A note on Chebyshev polynomials

New families of generating functions and identities concerning the Chebyshev polynomials are derived. It is shown that the proposed method allows the derivation of sum rules involving products of Chebyshev polynomials and addition theorems. The possiblity of extending the results to include gnerating functions involving products of Chebyshev and other polynomials is finally analyzed.

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Sunto Si derivano nuove famiglie di funzioni generatrici e di identità relative ai polinomi di Chebyshev. Si dimostra che il metodo proposto permette la derivazione di regole di somma relative a prodotti di polinomi di Chebyshev e teoremi di addizione. La possibilità di estendere i risultati includendos funzioni generatrici di prodotti di polinomi di Chebyshev ed altri polinomi è infine analizzata.

     

A note on linear permutation polynomials

Kai Zhou (2008) [8] gave an explicit representation of the class of linear permutation polynomials and computed the number of them. In this paper, we give a simple proof of the above results.     

A note on constructing permutation polynomials

Let H be a subgroup of the multiplicative group of a finite field. In this note we give a method for constructing permutation polynomials over the field using a bijective map from H to a coset of H. A similar, but inequivalent, method for lifting permutation behaviour of a polynomial to an extension field is also given.     

A note on specializations of Grothendieck polynomials

Buch and Rimányi proved a formula for a specialization of double Grothendieck polynomials based on the Yang–Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by Yong and Woo by constructing a Gröbner basis for the Kazhdan–Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators.     

A note on flow polynomials of graphs

Using the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by providing a general situation in which the integral flow polynomial is a multiple of the modular flow polynomial.     

A note on the coefficients of inverse polynomials

We describe some relations on the coefficients of a polynomial in terms of the map that induces and use them to characterize the coefficients of the inverse polynomials of some special classes of permutation polynomials.     

A note on the Dirichlet characters of polynomials

The main purpose of this paper is using the analytic methods and the properties of character sums to study the computational problem of several kind character sums of polynomials mod q, an odd square-full number, and give two interesting identities for them.     

A note on a family of two-variable polynomials

The main object of this paper is to construct a two-variable analogue of Jacobi polynomials and to give some properties of these polynomials. We show that these polynomials are orthogonal, then we obtain various recurrence formulas for them. Furthermore, we give some integral representations for these polynomials.     

A note on Hermite polynomials of several variables

The present paper deals with an extension of certain results obtained by Burchnall for Hermite polynomials to similar results for Hermite polynomials of several variables.     

A note on null subspaces of odd polynomials

Mathematical Notes - We obtain a generalization of the Aron-Hájek theorem about the null subspaces of homogeneous odd polynomials (in several variables) to the case of arbitrary odd polynomials.