  The zeros of D'Arcais polynomials, also known as Nekrasov--Okounkov
polynomials, dictate the vanishing of the Fourier coefficients of powers of the
Dedekind functions. These polynomials satisfy difference equations of
hereditary type with non-constant coefficients. We relate the D'Arcais
polynomials to polynomials satisying a Volterra difference equation of
convolution type. We obtain results on the transfer of the location of the
zeros.
  As an application, we obtain an identity between Chebyshev polynomials of the
second kind and $1$-associated Laguerre polynomials. We obtain a new version of
the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.
