A characterization of the exponential symmetric Sheffer sequences. Issue 12 (1st December 2020) - Wang, Weiping; Zhang, Ke
ABSTRACT: A polynomial sequence ( s n ( x ) ) n ∈ N is symmetric, or self-dual, if s n ( m ) = s m ( n ) for all m, n = 0, 1, 2, … . In this paper, we give the characterization of the exponential symmetric Sheffer sequences with weight sequences, and show that all the symmetric Sheffer sequences with weight ( λ n ), ( ⟨ λ ⟩ n ) or ( ( λ ) n ) are in fact constant multiples of the Charlier polynomial sequence, the Meixner polynomial sequence, or the Krawtchouk polynomial sequence, respectively, where ⟨ λ ⟩ n are the rising factorials, and ( λ ) n are the falling factorials.