Survey of binary Krawtchouk polynomials (Q2717206)

Binary Krawtchouk polynomials \(K_k^n(x)=\sum_{j=0}^k(-1)^j \binom{x}{j}\binom{n-x}{k-j}\) [see \textit{M. Krawtchouk}, Comptes Rendus 189,620-622 (1929; JFM 55.0799.01)] used in combinatorial problems related to the binary Hamming association schemes [see \textit{V. Levenshtein}, IEEE Trans. Inf. Theory 41, 1303-1321 (1995; Zbl 0836.94025)] are considered. In the paper it is attempted to review what is known about such polynomials. In the ``Definitions and Basic Properties'' section explicit expressions, integral formulas, moment sums, recurrence, symmetry and orthogonality relations are given. In the ``Connection to Other Polynomials'' section the connection of Krawtchouk polynomials to Lloyd polynomials, Hahn polynomials, Chebyshev polynomials and Hermite polynomials are considered. Some modular properties such as \(K_k^n(x)\equiv K_k^{n-q}(x)+ K_{k-q}^{n-q}(x) (\mod p)\), where \(p\) is a prime and \(q=p^s\) are given too. In section ``Zeros of Krawtchouk Polynomials'' some estimations of the smallest root and upper bounds are derived. A certain attention is devoted to existence of non-integer roots. For example, for arbitrary \(k\) the number of integer roots is at most \((9n/4)^{2/3}\). Some bounds on Krawtchouk polynomials and asymptotics for values of Krawtchouk polynomials in the oscillatory region (i.e. where the roots exist) are stated.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00079].