Carlitz Polynomials Research Articles

Notes on Askey–Roy integral and certain generating functions for [formula omitted]-polynomials

In this paper, generalizations of Askey–Roy integral are given by the technique of transformation. In addition, a recurring formula for Askey–Roy integral and an integral involving Cauchy polynomial are deduced by Askey–Roy integral. Besides, a new transformation identity is gained by Askey–Roy integral. Moreover, a certain generating function for generalized Al-Salam–Carlitz polynomials is obtained by Askey–Roy integral. At last, two generalizations of U(n+1) type generating function for generalized Al-Salam–Carlitz polynomials are derived by Askey–Roy integral.

An operator approach to the Al-Salam–Carlitz polynomials

We present an operator approach to Rogers-type formulas and Mehler’s formula for the Al-Salam–Carlitz polynomials Un(x,y,a;q). By using the q-exponential operator, we obtain a Rogers-type formula, which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler’s formula due to Al-Salam and Carlitz [“Some orthogonal q-polynomials,” Math. Nachr. 30, 47 (1965)]. By means of the Cauchy companion augmentation operator, we obtain an equivalent form of Mehler’s formula. We also give several identities on the generating functions for products of the Al-Salam–Carlitz polynomials, which are extensions of the formulas for the Rogers–Szegö polynomials.

A recurring [formula omitted]-integral formula

In this paper, we use the q -Chu–Vandermonde formula to derive a recurring q -integral formula. Some applications of the recurring q -integral formula are also given, which include deriving the 3 ϕ 2 representation of the Al-Salam–Carlitz polynomials and the extension of the terminating Sears’ 3 ϕ 2 transformation formula.

Formula omitted]-integral representation of the Al-Salam–Carlitz polynomials

We use the Andrews–Askey integral and the Leibniz rule for the q -difference operator to give the q -integral representation of the Al-Salam–Carlitz polynomials, which includes the q -integral representation of the Rogers–Szegö polynomials and the q -integral representation of the q -Hermite polynomials as special cases.

Dedekind–Carlitz polynomials as lattice-point enumerators in rational polyhedra

We study higher-dimensional analogs of the Dedekind–Carlitz polynomials $${c(u,v;a,b): = \sum\limits_{k = 1}^{b - 1} {u^{\left[ {\frac{{ka}}{b}} \right]} } v^{k - 1}}$$ , where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law $${(v - 1)\,{\rm c}\,(u, v; a, b) + (u - 1)\,{\rm c}\,(v, u; b, a) = u^{a-1} v^{b-1} - 1,}$$ from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind–Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for Dedekind–Carlitz polynomials, a characterization of Dedekind–Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell–Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes.

Carlitz and the general 3φ2

In a letter dated March 3, 1971, L. Carlitz defined a sequence of polynomials, Φ n (a,b; x, y; z), generalizing the Al-Salam & Carlitz polynomials, but closely related thereto. He concluded the letter by stating: “It would be of interest to find properties of Φ n (a, b; x, y; z) when all the parameters are free.” In this paper, we reproduce the Carlitz letter and show how a study of Carlitz’s polynomials leads to a clearer understanding of the general 3Φ2 (a, b, c; d; e; q, z).

Some Basic 𝑑-Orthogonal Polynomial Sets

Abstract The purpose of this paper is to study the class of polynomial sets which are at the same time 𝑑-orthogonal and 𝑞-Appell. By a linear change of variable, the resulting set reduces to 𝑞-Al-Salam–Carlitz polynomials, for 𝑑 = 1. Various properties of the obtained polynomials are singled out: a generating function, a recurrence relation of order 𝑑 + 1. We also explicitly express a 𝑑-dimensional functional for which the d-orthogonality holds.

Umbral calculus in positive characteristic

An umbral calculus over local fields of positive characteristic is developed on the basis of a relation of binomial type satisfied by the Carlitz polynomials. Orthonormal bases in the space of continuous F q -linear functions are constructed.

Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials

We study operator pencils on generators of the Lie algebras sl2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2F1 and degenerate 1F1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen–van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi–Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered.

Cn-Functions over completions of [formula omitted] at finite places of [formula omitted

It is known from Wagner's theorem (Acta Arith. 17 (1971) 389) that a continuous function f from O to K can be expanded using Carlitz polynomials G j ( x): f(x)=∑ j=0 ∞ a j G j(x) , with O equal to the completion of F r[T] at a prime ℘ and K the quotient field of O . Using the definition of a C n -function by Schikhof (Ultra-Metric Calculus: An Introduction to p-adic Analysis, Cambridge Studies in Advanced Mathematics, Vol. 4, Cambridge University Press, Cambridge, 1984), we prove in this paper that f( x) is a C n -function if and only if lim j→∞ |a j| j n=0 . An alternative approach to Wagner's result is also given in the paper.

Stieltjes transforms of a class of probability measures

Let the probability measures μN,N=2,3,… be defined by μN({λk,N})=1/N,μN(A)=0 for λk,N∉A, where λk,N,k=1,2,…,N are the zeros of the orthogonal polynomial PN+1(x), which is obtained recursively by P0(x)=0,P1(x)=1,anPn+1(x)+an−1Pn−1(x)+bnPn(x)=xPn(x). Conditions on an and bn are found such that the sequence μN,N=2,3,… converges weakly to the Dirac measure at the point zero. This is achieved through the convergence of the sequence of Stieltjes transforms ∫−∞∞dμN(t)/(λ−t) to the function 1/λ. Typical examples are the Al-Salam and Carlitz polynomials, the Wall polynomials, the Lommel polynomials and the Tricomi–Carlitz polynomials.

Convergent sequences of orthogonal polynomials

are precisely the spectral points of and ifthe moment problem is indeterminate, they are the spectral points of an extremal solution. For theabove, see [1].Examples of this phenomenon are provided by the Al-Salam and Carlitz q-polynomials and byStieltjes–Carlitz polynomials [2]. However, in all of these examples, we actually have

A Comparison of the Carlitz and Digit Derivatives Bases in Function Field Arithmetic

We compare several properties and constructions of the Carlitz polynomials and digit derivatives for continuous functions on Fq[[T]]. In particular, we show a close relation between them as orthonormal bases. Moreover, parallel to Carlitz's coefficient formula, we give the closed formula for the expansion coefficients in terms of the digit derivatives.

Isomorphisms of type A affine Hecke algebras and multivariable orthogonal polynomials

We examine two isomorphisms between affine Hecke algebras of type A associated with parameters q-1, t-1 and q, t. One of them maps the non-symmetric Macdonald polynomials Eη(x;q-1,t-1) onto Eη(x;q,t), while the other maps them onto non-symmetric analogues of the multivariable Al-Salam & Carlitz polynomials. Using the properties of Eη(x;q-1,t-1), the corresponding properties of these latter polynomials can then be elucidated.

Multivariable Al-Salam & Carlitz Polynomials Associated with the TypeA q-Dunkl Kernel

The Al–Salam & Carlitz polynomials are q–generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the q–analogue of the type–A Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with respect to a certain Jackson integral inner product. This inner product is normalized by deriving a q–analogue of the Mehta integral, and the corresponding normalization of the multivariable Al–Salam & Carlitz polynomials is derived from a Pieri–type formula. Various other special properties of the polynomials are also presented, including their relationship to the shifted Macdonald polynomials and the big–q Jacobi polynomials.

The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

We study the zeros of orthogonal polynomials pn, N, n=0, 1, …, that are generated by recurrence coefficients an, N and bn, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio n/N converges, we show that the polynomials pn, N have an asymptotic zero distribution as n/N tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distri- butions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi–Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes–Wigert polynomials, and Lommel polynomials.

Locally Analytic Functions over Completions of [formula omitted]r[U

It is well known that a continuous functionf:Zp→Qpcan be expanded by Mahler's basis: [formula] withan→0. Amice (Bull. Soc. Math. France92(1964), 117–180) has established conditions on the coefficientsanfor the functionfto be locally analytic, as well as more general results whenZpis replaced by some compact subset of a local field. We study the function field case in this paper. The function field analogue of Mahler's basis is the Carlitz polynomials, and the corre- sponding result for continuous functions has already been established by Wagner (Acta Arith.17(1971), 389–406). We show that the conditions for a continuous function to be locally analytic in the function field case are completely similar to theQpcase. An application to using integral calculus to analytically continue characteristicp-valuedL-series is briefly mentioned at the end of the paper.

Harmonic Oscillator in Characteristic p

We construct an irreducible representation of the canonical commutation relations by operators on a certain Banach space over a local field of characteristic p. The Carlitz polynomials forming the basis of the space are shown to be the counterparts of the Hermite functions for this situation. The analogues of coherent states are related to the Carlitz exponential.

On Combinatorics of Al-Salam Carlitz Polynomials

A new combinatorial interpretation of the moments of Al-Salam Carlitz polynomials as ‘striped’ skew-shapes is used to explain the cancellation in the moments of Viennot theory for these polynomials.

On the Asymptotics of the Tricomi–Carlitz Polynomials and Their Zero Distribution (I)

The asymptotic behavior of the Tricomi–Carlitz polynomials in the complex plane is established.MSC codes33A6541A60Keywordsorthogonal polynomialszero distributionStieltjes transformuniform convergencetheorem of Grommer and Hamburger