Diego Dominici
We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials.
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The generating function of Stieltjes-Carlitz polynomials is a solution of Heun's differential equation and using this relation Carlitz was the first to get exact closed forms for some Heun functions. Similarly the associated Stieltjes-Carlitz polynomials lead to a new differential equation which we call associated Heun. Thanks to the link with orthogonal polynomials we are able to deduce two integral relations connecting associated Heun functions with different parameters and to exhibit the set of associated Heun functions which generalize Carlitz's. Part of these results were used by the auth…
On the Carlitz-Mehler formula for Hermite polynomials
本文给出Hermite多项式Carlitz-Mehler公式的进一步推广的直接证明，扩展了Carlitz和Srivastava等人的工作。
In this paper, by making use of the familiar $q$-difference operators $D_q$ and $D_{q^{-1}}$, we first introduce two homogeneous $q$-difference operators $\mathbb{T}({\bf a},{\bf b},cD_q)$ and $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$, which turn out to be suitable for dealing with the families of the generalized Al-Salam-Carlitz $q$-polynomials $ϕ_n^{({\bf a},{\bf b})}(x,y|q)$ and $ψ_n^{({\bf a},{\bf b})}(x,y|q)$. We then apply each of these two homogeneous $q$-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas and the Srivas…
According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and Euler polynomials. Furthermore, we find new formulas for the Carlitz's (Carlitz) and Howard's (Howard2) finite sums. Finally, we evaluate several series in closed forms, one of which has the coefficients include values of the Riemann zeta function. Moreover, we calculate some integrals in terms of generalized geometric polynomials.
Explicit solutions for the three-term recurrence satisfied by associated continuous dual $q$-Hahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The absolutely continuous component of the spectral measure is obtained. Eleven limit cases are discussed in some detail. These include associated big $q$-Laguerre , associated Wall, associated Al-Salam-Chihara, associated Al-Salam-Carlitz I, and associated continuous $q$-Hermite polynomials.
We state and prove the $q$-extension of a result due to Johnston and Jordaan (cf. \cite{Johnston-2015}) and make use of this result, the orthogonality of $q$-Laguerre, little $q$-Jacobi, $q$-Meixner and Al-Salam-Carlitz I polynomials as well as contiguous relations satisfied by the polynomials, to establish the quasi-orthogonality of certain $_2ϕ_2$ and $_3ϕ_2$ polynomials. The location and interlacing properties of the real zeros of these quasi-orthogonal polynomials are studied. Interlacing properties of the zeros of $q$-Laguerre quasi-orthogonal polynomials $L_n^{(δ)}(z;q)$ when $-2<δ<-1$ w…
Elements of the q-Askey scheme in the algebra of symmetric functions
本研究在对称函数代数中构造多变量q-Racah对称函数，推导其可退化为big q-Jacobi等三类对称函数，证明四类函数均关于无穷点构形空间上的测度正交，且测度间的极限转移与函数退化一致，丰富了q-Askey概型的多变量对称函数理论。
Some observations on the Wu-Sprung potential.
A class of solenoids is considered, including some aspects in n (topological) dimensions, where one basically gets some fractal versions of tori.
These informal notes briefly discuss various aspects of Cantor sets.
A basic class of constructions is considered, in connection with bilipschitz mappings in particular.
This paper briefly discusses some questions with analytical and topological aspects, with hopefully some useful references on both sides.
Some aspects of Cauchy integrals on sets with dimension larger than 1 are briefly discussed.
This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.
A basic family of solenoids is discussed, especially from the point of view of analysis on metric spaces.
Here we consider a few topics related to Lipschitz classes for functions and curves in metric spaces.
In this notice, we revisit the recent work [1] of Jung Yoog Kang and Tai Sup about special polynomials with exponential distribution in order to state some improvements and get new proofs for results therein.
The "potentials" being considered are analogues of classical Riesz potentials of order 1, and the idea is to look at how they might map L^p spaces into Sobolev spaces in various settings.
The first section of this modest survey reviews some basic notions and describes some families of examples, and the second section briefly indicates some general aspects of analysis on metric spaces. The remaining three sections are concerned with some particular situations involving sub-Riemannian geometry, hyperbolic groups, and p-adic numbers.
We argue that another proof by Trimeche of the geometrical form of the Paley-Wiener theorems for the Dunkl transform is not correct.
The property 4 in Proposition 2.3 from the paper "Some remarks on Davie's uniqueness theorem" is replaced with a weaker assertion which is sufficient for the proof of the main results. Technical details and improvements are given.
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