The term "poweroid" has at least two meanings. Sheffer sequences are sometimes called poweroids (Steffensen 1941, Shiu 1982, Di Bucchianico
and Loeb 2000). Jackway and Deriche (1996) and Jackway (2002) use the term to mean
a function of the form
The case
corresponds to the paraboloid and the case is sometimes called a quartoid
(Jackway and Deriche 1996).
Di Bucchianico, A. and Loeb, D. "A Selected Survey of Umbral Calculus." Electronic J. Combinatorics Dynamical Survey DS3,
1-34, April 2000. http://www.combinatorics.org/Surveys/#DS3.Jackway,
P. T. "Partial Orderings and Scale-Spaces with Monotonicity of Features."
In Proceedings
of ISMM2002 (Ed. H. Talbot and R. Beare). CSIRO Publishing, pp. 277-282,
2002.Jackway, P. T. and Deriche, M. "Scale-Space Properties
of the Multiscale Morphological Dilation-Erosion." IEEE Trans. Patterns Anal.
Mach. Intell.18, 38-51, 1996.Shiu, E. S. W. "Steffensen's
Poweroids." Scand. Actuar. J.2, 123-128, 1982.Steffensen,
J. F. "The Poweroid, an Extension of the Mathematical Notion of Power."
Acta Math.73, 333-366, 1941.
corresponds to the paraboloid and the case 
is sometimes called a quartoid
(Jackway and Deriche 1996).
