On totally positive sequences and functions

Let a > 0 be a fixed number. A function f : R → R is said to be a-shift-generating (a-SG) if for every x ∈ R , { f ( a n − x ) } n = − ∞ ∞ is a totally positive sequence and it does not coincide with a sequence of the form { A λ n } n = − ∞ ∞ , where A ⩾ 0 and λ > 0 . In this paper, we describe all a-SG functions and obtain a new characterization of totally positive functions in the terms of a-SG functions. In addition, using characteristic properties of a-SG functions, we generalize the famous Jacobian identity in theory of elliptic functions.