## 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.)

JFM 55.0799.01
Full Text:

### 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
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