  Motivated by the study of the distribution of zeros of generalized
Bessel-type functions, the principal goal of this paper is to identify new
research directions in the theory of multiplier sequences. The investigations
focus on multiplier sequences interpolated by functions which are not entire
and sums, averages and parametrized families of multiplier sequences. The main
results include (i) the development of a `logarithmic' multiplier sequence and
(ii) several integral representations of a generalized Bessel-type function
utilizing some ideas of G. H. Hardy and L. V. Ostrovskii. The explorations and
analysis, augmented throughout the paper by a plethora of examples, led to a
number of conjectures and intriguing open problems.
