Derivations and identities for Kravchuk polynomials. (English. Ukrainian original) Zbl 1314.42025

Ukr. Math. J. 65, No. 12, 1755-1773 (2014); translation from Ukr. Mat. Zh. 65, No. 12, 1587-1603 (2013).
This paper gives quite a number of ‘derivations’ (a linear mapping of the algebra \(\mathbb{Q}(x_0,x_1,\ldots,x_n)\) of rational polynomials in \((n+1)\) variables, satisfying the Leibniz rule) and identities for Kravchuk polynomials.
These polynomials were introduced by M. Krawtchouk [C. R. Acad. Sci., Paris 189, 620–622 (1929; JFM 55.0799.01)] and are given by \[ K_n(x,a)=\sum_{i=0}^n\,(-1)^i{x\choose i}{a-x\choose n-i}. \] In the paper under review a general method is proposed to find identities of the form \[ P(K_0(x,a),K_1(x,a),\ldots,K_n(x,a))=\varphi_1(a)\text{ or }\varphi_2(x), \] where \(P(x_0,x_1,\ldots,x_n)\) is a polynomial in \(n+1\) variables and \(\varphi_1,\varphi_2\) are polynomials in one variable.
The layout of the paper is as follows:
§1: Introduction (5 pages).
Definitions and discussion of methods.
§2: Locally nilpotent derivations and Kravchuk diffeomorphisms (8 pages).
§3: Permutable mappings for the Weizenböck and Kravchuk derivations (6 pages).
References (9 items).

The paper contains three hypotheses following from the explicit calculations for small values of \(n\). \[ \begin{align*}{(1)\quad\sum_{i=0}^n\,K_i(&x,a)\sum_{k=0}^{n-i}\,{(-1)^k\over k!}K_1(x,a)^kS^{(k)}(n-i)= \cr & =0\ (n\text{ odd}),\cr &=(-1)^m(2m-1)!!a(a-2)(a-4)\cdots (a-2(m-1))\ (n=2m), \text{where}\cr &S^{(k)}(n)=\sum_{m=k}^n\,{n-1\choose m-1}{2^mk!\over m!}s(m,k),\quad s(m,k)\text{ Stirling numbers of the first kind}.\cr (2)\quad\sum_{i=0}^n\,K_i(&x,a)\sum_{k=0}^{n-i}\,{(-1)^k\over(n-i)!}K_1(x,a)^ks(n-i,k)=\cr &=0\text{ if }n\text{ is odd},\cr &=(-1)^m{x\choose m}\text{ if }n=2m.\cr (3)\quad \varphi_1(a)=&(-1)^{n(n+1)/2}\prod_{i=0}^n\,i!\prod_{i=0}^{n-2}\,(a+i)^{n-1-i},\ \varphi_2(x)= (-1)^{n(n+1)/2}\prod_{i=0}^n\,2^i i!\prod_{i=0}^{n-2}\,(x-i)^{n-1-i}.}\end{align*} \]


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)


JFM 55.0799.01
Full Text: DOI arXiv


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