  The influence of a variable is an important concept in the analysis of
Boolean functions. The more general notion of influence of a set of variables
on a Boolean function has four separate definitions in the literature. In the
present work, we introduce a new definition of influence of a set of variables
which is based on the auto-correlation function and develop its basic theory.
Among the new results that we obtain are generalisations of the Poincar\'e
inequality and the edge expansion property of the influence of a single
variable. Further, we obtain new characterisations of resilient and bent
functions using the notion of influence. We show that the previous definition
of influence due to Fischer et. al. (2002) and Blais (2009) is half the value
of the auto-correlation based influence that we introduce. Regarding the other
prior notions of influence, we make a detailed study of these and show that
each of these definitions do not satisfy one or more desirable properties that
a notion of influence may be expected to satisfy.
