For all natural numbers \(n\) and \(x\), the degree-zero binary Krawtchouk polynomial is identically \(1\); that is,

Let \(n\) and \(j\) be natural numbers with \(j \le n\). Then the binary Krawtchouk polynomial \(K_j^{(n)}\) evaluated at \(0\) equals the binomial coefficient \(\binom {n}{j}\), that is,

Let \(n\) and \(x\) be natural numbers with \(x \le n\). Then the first binary Krawtchouk polynomial at \(x\) satisfies

Let \(n\) and \(x\) be natural numbers with \(x \le n\), and let \(K_j^{(n)}(x)\) denote the binary Krawtchouk polynomial. Then for every real \(z\),

Fix a natural number \(n\), and let \(r, s\) be natural numbers with \(r \le n\) and \(s \le n\). Then the binary Krawtchouk polynomials satisfy

that is, they are orthogonal with respect to the weights \(\binom {n}{x}\) on \(\{ 0, 1, \dots , n\} \).

Let \(n\), \(j\), and \(x\) be natural numbers with \(1 \le j\), \(j + 1 \le n\), and \(x \le n\). Then the binary Krawtchouk polynomials \(K_j^{(n)}\) satisfy the three-term recurrence

Fix a natural number \(n\), and let \(j, x\) be natural numbers with \(j \le n\) and \(x \le n\). Then the Krawtchouk polynomial \(K_j^{(n)}\), evaluated at the real number \(x\), equals the alternating sum

Fix naturals \(n\) and \(d\), and let \(F_0, F_1, \dots , F_n\) be real coefficients defining the function \(F(x) = \sum _{j=0}^{n} F_j\, K_j^{(n)}(x)\). Suppose that \(F_0 \gt 0\), that \(F_j \ge 0\) for every \(j\) with \(1 \le j \le n\), and that \(F(x) \le 0\) for every integer \(x\) with \(d \le x \le n\). Then the maximum size of a binary code of length \(n\) and minimum distance \(d\) satisfies

Let \(n, t\) be natural numbers with \(t + 1 \le n\), and let \(a, x\) be real numbers with \(a \ne x\). Then there is a positive constant \(c \gt 0\) for which the truncated Christoffel–Darboux kernel satisfies

where \(K_j^{(n)}\) denotes the \(j\)-th Krawtchouk polynomial.

Let \(n\) and \(t\) be natural numbers with \(t + 1 \le n\), and let \(K_j^{(n)}\) denote the degree-\(j\) Krawtchouk polynomial associated with \(n\). Suppose \(a\) is a real number lying strictly below every real zero of \(K_t^{(n)}\): that is, \(a \lt \xi \) whenever \(K_t^{(n)}(\xi ) = 0\). Then \(K_j^{(n)}(a) \gt 0\) for every \(j\) with \(j \le t\).

Fix natural numbers \(t\) and \(n\) with \(t + 1 \le n\), and let \(a\) be a real number lying strictly to the left of every real zero of the Krawtchouk polynomial \(K_t^{(n)}\); that is, \(a \lt \xi \) whenever \(K_t^{(n)}(\xi ) = 0\). Then there is a real threshold \(\eta \) such that the truncated Christoffel–Darboux kernel satisfies \(\Phi _{n,t}(a, x) \le 0\) for every integer \(x\) with \(\eta \le x \le n\).

Fix natural numbers \(n\), \(t\), and \(d\) with \(t + 1 \le n\), and let \(a\) be a real number lying strictly below every real zero of the Krawtchouk polynomial \(K_t^{(n)}\). Suppose that the truncated Christoffel–Darboux kernel satisfies \(\Phi _{n,t}(a, x) \le 0\) for every integer \(x\) with \(d \le x \le n\). Then the maximum size of a binary code of length \(n\) and minimum distance \(d\) is bounded by

Fix \(\tau \) with \(0 \lt \tau \lt 1/2\), and let \((t_n)\) be a sequence of natural numbers with \(t_n/n \to \tau \). Then there is a real sequence \((a_n)\) with the following three properties. First, for all sufficiently large \(n\), the number \(a_n\) lies strictly below every real zero of the Krawtchouk polynomial \(K_{t_n}^{(n)}\); that is, \(K_{t_n}^{(n)}(\xi ) = 0\) implies \(a_n \lt \xi \). Second, \(a_n/n \to \tfrac 12 - \sqrt{\tau (1-\tau )}\). Third, for all sufficiently large \(n\), the truncated Christoffel–Darboux kernel satisfies \(\Phi _{n,t_n}(a_n, x) \le 0\) for every integer \(x\) with \(\big\lfloor (\tfrac 12 - \sqrt{\tau (1-\tau )})\, n \big\rfloor \le x \le n\).

Fix \(0 \lt \tau \lt 1/2\), and let \((t_n)\) be a sequence of natural numbers and \((a_n)\) a sequence of reals such that \(t_n/n \to \tau \) and, for all sufficiently large \(n\), the point \(a_n\) lies strictly below every real zero of the Krawtchouk polynomial \(K_{t_n}^{(n)}\). Then the truncated Christoffel–Darboux kernel evaluated at \((a_n, 0)\) grows at rate at most \(H(\tau )\), in the sense that

where \(H\) is the binary entropy function in base \(2\).

Fix a real number \(\tau \) with \(0 \le \tau \le \tfrac 12\). Then the normalized base-\(2\) logarithm of the lower binomial tail sum converges to the binary entropy of \(\tau \):

Let \(\delta \) be a real number with \(0 \le \delta \le \tfrac 12\), and define \(\tau = \tfrac 12 - \sqrt{\delta (1-\delta )}\). Then \(\tau \) again lies in \([0,\tfrac 12]\) and applying the same rule to it recovers \(\delta \), that is, \(\tfrac 12 - \sqrt{\tau (1-\tau )} = \delta \). Hence the map \(x \mapsto \tfrac 12 - \sqrt{x(1-x)}\) is an involution of the interval \([0,\tfrac 12]\).

Let \(H(x) = -x\log _2 x - (1-x)\log _2(1-x)\) be the binary entropy function in base \(2\), under the convention \(0\log _2 0 = 0\). Then \(H(0) = 0\).

The binary entropy function \(H\) attains the value \(1\) at the point \(1/2\); that is, \(H(1/2) = 1\).

For every real \(\delta \) with \(0 \le \delta \le 1/2\), the asymptotic binary code rate satisfies

where \(H\) is the base-\(2\) binary entropy function.