On combinatorial properties and the zero distribution of certain Sheffer sequences

The zero locus and some combinatorial properties of certain exponential Sheffer sequences

“…The main result of the paper concerning the zeros of certain exponential Sheffer sequences is the following theorem. We point out that this result is stronger than the results found in a string of recent works on the zero distribution of various polynomial sequences (see for example [2], [4], [5], [6]) in that the present paper provides the exact curve on which the zeros of the P m s lie for all m, not just for m 1. We are able to establish the main result for all P m due to a simple differential recurrence relation the P m s must satisfy (see the opening discussion of Section 2), essentially identifying the the shift operator P m ∆ −→ P m−1 as scaled differentiation -a hyperbolicity preserving linear operator.…”

“…The next proposition provides a key element in our proof of the main result, by establishing the existence of a curve, along which we find a suitable asymptotic expansion of the integral representing the polynomials in question. Proposition 6 is the analog of Proposition 24 in [2], but is appreciably simpler to establish. The nature (and singularities) of the generating function in [2] necessitated a lengthy discussion on the topology of the level sets of Re φ(z(y), s) when extending the curve from a local piece around ζ(s) to (complex) infinity.…”

“…Proposition 6 is the analog of Proposition 24 in [2], but is appreciably simpler to establish. The nature (and singularities) of the generating function in [2] necessitated a lengthy discussion on the topology of the level sets of Re φ(z(y), s) when extending the curve from a local piece around ζ(s) to (complex) infinity. No such discussion is necessary in the current work, since the generating function of the Sheffer sequence under consideration has no singularities.…”

“…The following lemma follows immediately from [2,3,7]. The basic idea is that marked labels annihilate the non-marked labels with the same number at the same level.…”

“…The current work can be rightfully considered as a sequel to a recent paper of the the first, second and fourth authors regarding Sheffer sequences, their zeros, and some combinatorial properties (see [2]). In that work we give a brief overview of Sheffer sequences and their history, as well as a short description of some of the recent works concerning the study of the zero distribution of polynomial sequences satisfying various types of recurrence relations.…”