Zeros of generalized Krawtchouk polynomials (Q584473)

Let \(k_ n(x,q,N)\), \(n=Q,1,...,N\), be the finite sequel of the Krawtchouk polynomials having the generating function \((1+(q-1)z)^{N- x}(1-z)^ x.\) The authors give some elementary interlacing properties of the zeros of these polynomials; they establish some necessary conditions for such a polynomial to have an integral zero, give a non-trivial family of integral zeros, namely \(k_{2h}\) \((4h-1,2,8h+1)=0\) for any integer \(h\geq 1\) and conjecture that the number of non-trivial integral zeros of \(k_ n(x,2,M)\) for \(M\leq N\) is asymptotic to N/8. Finally, they show that a family of q-Krawtchouk polynomials never has integral zeros and classify the integral zeros of a second class of q-Krawtchouk polynomials.