On the zeros of certain Sheffer sequences and their cognate sequences

Abstract: Given a Sheffer sequence of polynomials, we introduce the notion of an associated sequence called the cognate sequence. We study the relationship between the zeros of this pair of associated sequences and show that in case of an Appell sequence, as well as a more general family of Sheffer sequences, the zeros of the members of each sequence (for large n) are either real, or lie on a line $\Re (z)=c$. In addition to finding the zero locus, we also find the limiting probability distribution function of such sequences.