Zeros transfer for recursively defined polynomials

Abstract

The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying 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.

Raising Series to A Power

“…The class of polynomials τ α (n) are called D' Arcais polynomials (refer (4) ). The search for reducibility criterions of these polynomials over the ring of integers is of special interest and one can see a lot of papers appearing in this direction (refer (2,(5)(6)(7) ), the main reason behind this search is that the non-vanishing of the polynomials at α = −24 for each n is equivalent to Lehmer's conjecture on Ramanujan's tau function which is still open. As the relations (3) and ( 4) are well-known we take the above derivation as an illustration, and proceed in similar fashion taking into account the other partition-generating functions.…”