Generalized Bernstein Polynomials

We introduce polynomials B ⁿ _i (x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B ⁿ _i (x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

q-Bernstein polynomials and their iterates

Let B_n( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials B_n( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {B_n( f,q;x)} to f(x) in the norm of C[0,1] has the order q⁻ⁿ (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B_n^j_n( f,q;x)}, where both n→∞ and j_n→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j_n→∞.     

On explicit factors of cyclotomic polynomials over finite fields

We study the explicit factorization of 2 ⁿ r-th cyclotomic polynomials over finite field \mathbbF_q{\mathbb{F}_q} where q, r are odd with (r, q) = 1. We show that all irreducible factors of 2 ⁿ r-th cyclotomic polynomials can be obtained easily from irreducible factors of cyclotomic polynomials of small orders. In particular, we obtain the explicit factorization of 2 ⁿ 5-th cyclotomic polynomials over finite fields and construct several classes of irreducible polynomials of degree 2 ^n–2 with fewer than 5 terms.     

CONVERGENCE RATE FOR ITERATES OF q-BERNSTEIN POLYNOMIALS

Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {Bn（., qn）} when n → ∞ and convergence rate of Bn（f,q;x） when f ∈ C^n-1 [0, 1], q → ∞ are also presented.     

A characterization of some q-orthogonal polynomials

We show that the only orthogonal polynomials satisfying a q-difference equation of the form π(x)D _q P _n (x) = (α _n x + β _n )P _n (x) + γ _n P _n−1(x) where π(x) is a polynomial of degree 2, are the Al-Salam Carlitz 1, little and big q-Laguerre, the little and big q-Jacobi, and the q-Bessel polynomials. This is a q-analog of the work carried out in [1]. 2000 Mathematics Subject Classification Primary—33C45, 33D45     

Properties of convergence for -Bernstein polynomials

In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q⁻¹,…,q⁻ⁿ⁺¹. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), q_n(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if lim_n→∞q_n=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].     

A Generalization of the Kostka–Foulkes Polynomials

Combinatorial objects called rigged configurations give rise to q-analogues of certain Littlewood–Richardson coefficients. The Kostka–Foulkes polynomials and two-column Macdonald–Kostka polynomials occur as special cases. Conjecturally these polynomials coincide with the Poincaré polynomials of isotypic components of certain graded GL(n)-modules supported in a nilpotent conjugacy class closure in gl(n).     

Rational Functions and Association Scheme Parameters

The parameters of metric, cometric, symmetric association schemes with q ± 1 (the same as the parameters of the underlying orthogonal polynomials) can be given in general by evaluating a single rational function of degree (4, 4) in the complex variable q ^j. But in all known examples, save the simple n-gons, these reduce to polynomials of degree at most 2 in q ^j with q an integer. One reason this occurs is that the rational function can have singularities at points which would determine some of the parameters. This paper deals with the case in which not all of the singularities are removable, thus giving some reason why the n-gons might naturally be the only exceptions to schemes with parameters being polynomials of degree at most 2 in q ^j , except possibly for schemes of very small diameter.     

Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for

In the paper, we discuss Voronovskaya-type theorem and saturation of convergence for q-Bernstein polynomials for arbitrary fixed q, 0<q<1. We give explicit formulas of Voronovskaya-type for the q-Bernstein polynomials for 0<q<1. If , we show that the rate of convergence for the q-Bernstein polynomials is o(qⁿ) if and only ifWe also prove that if f is convex on [0,1] or analytic on (-ε,1+ε) for some ε>0, then the rate of convergence for the q-Bernstein polynomials is o(qⁿ) if and only if f is linear.     

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL_n(q), SL_n(q), PGL_n(q) and PSL_n(q), respectively. Using conjugacy classes of elements in GL_n(q) in terms of irreducible polynomials over the finite field GF(q) we demonstrate how the set of order elements in GL_n(q) can be obtained. This will help to find the order of elements in the groups SL_n(q), PGL_n(q) and PSL_n(q). We also show an upper bound for the order of elements in SL_n(q).     

A numerical investigation on the structure of the roots of q-Genocchi polynomials

One purpose of this paper is to introduce the q-Genocchi polynomials, G _n,q (x). Next we display the shape of q-Genocchi polynomials. Finally, we investigate the zeros of the q-Genocchi polynomials.     

A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

Let F _q[X] denote a polynomial ring over a finite field F _q with q elements. Let 𝒫_n be the set of monic polynomials over F _q of degree n. Assuming that each of the qⁿ possible monic polynomials in 𝒫_n is equally likely, we give a complete characterization of the limiting behavior of P(Ω_n=m) as n→∞ by a uniform asymptotic formula valid for m≥1 and n−m→∞, where Ω_n represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in 𝒫_n. The distribution of Ω_n is essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q⁻¹. Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional behavior being essentially a parabolic cylinder function (or some linear combinations of the standard normal law and its iterated integrals). As applications of this uniform asymptotic formula, we derive most known results concerning P(Ω_n=m) and present many new ones like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to Rényi's problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 17–47, 1998     

q-Laguerre polynomials and related q-partial differential equations

In this paper, we define two homogenous q-Laguerre polynomials, by introducing a modified q-differential operator, we prove that an analytic function can be expanded in terms of the q-Laguerre polynomials if and only if the function satisfies certain q-partial differential equations. Using this main result, we derive the generating functions, bilinear generating functions and mixed generating functions for the q-Laguerre polynomials and generalized q-Hahn polynomials. Cigler’s polynomials and its generating functions discussed in [J. Cao, D.-W. Niu, A note on q -difference equations for Cigler’s polynomials, J. Difference Equ. Appl. 22 (2016), 1880–1892.] are generalized. At last, we obtain an q-integral identity involving q-Laguerre polynomials. These applications indicate that the q-partial differential equation is an effective tool in studying q-Laguerre polynomials.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

The correlation functions of vertex operators and Macdonald polynomials

The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n = 1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.     

Minimal cyclic codes of length pq

Explicit expressions for all the 3n+2 primitive idempotents in the ring R_pⁿq=GF(ℓ)[x]/(x^pnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pⁿ and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pⁿq over GF(ℓ) are also discussed.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

A constructive approach to Noether's problem

A general method is developed to attack Noether's Problem constructively by trying to find minimal bases consisting of rational invariants which are quotients of polynomials of small degrees. This approach turns out to be successful for many small groups and for most of the classical groups with their natural representations. The applications include affirmative answers to Noether's Problem for the conformal symplectic groups CSp _2n (q), for the simple subgroups Ω _n (q) of the orthogonal groups forn andq odd, for some other subgroups of orthogonal groups and for the special unitary groups SU _n (q ²). The author was supported by the Graduate College “Modelling and Scientific Computing in Mathematics and Science” during this work     

The tree method for multidimensional <Emphasis Type="Italic">q</Emphasis>-Hahn and <Emphasis Type="Italic">q</Emphasis>-Racah polynomials

We develop a tree method for multidimensional q-Hahn polynomials. We define them as eigenfunctions of a multidimensional q-difference operator and we use the factorization of this operator as a key tool. Then we define multidimensional q-Racah polynomials as the connection coefficients between different bases of q-Hahn polynomials. We show that our multidimensional q-Racah polynomials may be expressed as product of ordinary one-dimensional q-Racah polynomial by means of a suitable sequence of transplantations of edges of the trees. Our paper is inspired to the classical tree methods in the theory of Clebsch–Gordan coefficients and of hyperspherical coordinates. It is based on previous work of Dunkl, who considered two-dimensional q-Hahn polynomials. It is also related to a recent paper of Gasper and Rahman: we show that their multidimensional q-Racah polynomials correspond to a particular case of our construction.