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

MSC:

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.)

Citations:

JFM 55.0799.01
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Full Text: DOI arXiv

References:

[1] Krawtchouk, M, Sur une generalisation des polynomes d’hermite, C. R. Acad. Sci., 189, 620-622, (1929) · JFM 55.0799.01
[2] A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin (1991). · Zbl 0743.33001
[3] Pryzva, GY, Kravchuk orthogonal polynomials, Ukr. Math. J., 44, 792-800, (1992) · Zbl 0783.33006
[4] Bedratyuk, L, Semiinvariants of binary forms and identities for Bernoulli, Euler, and Hermite polynomials, Acta Arithm., 151, 361-376, (2012) · Zbl 1257.11020
[5] Bedratyuk, L, Derivations and identities for Fibonacci and Lucas polynomials, Fibonacci Quart., 51, 351-366, (2013) · Zbl 1352.11030
[6] Krasikov, I; Litsyn, S, On the integral zeros of krawtchouk polynomials, J. Combin. Theory, Ser. A, 74, 71-99, (1996) · Zbl 0853.33008
[7] Koepf, W, Identities for families of orthogonal polynomials and special functions, Integral Transform. Spec. Funct., 5, 69-102, (1997) · Zbl 0895.33002
[8] O. Glenn, A Treatise on Theory of Invariants, Ginn & Co., Boston (1915). · JFM 45.0240.01
[9] A. Nowicki, Polynomial Derivations and Their Ring of Constants, Uniw. Mikolaja Kopernika, Torun (1994). · Zbl 1236.13023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.