  Many combinatorial and other number triangles are solutions of recurrences of
the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining
recurrences are investigated analytically. They are acted on by a
transformation group generated by two involutions: a left-right reflection and
an upper binomial transformation, acting row-wise. The group also acts on the
bivariate exponential generating function (EGF) of the triangle. By the method
of characteristics, the EGF of any GKP triangle has an implicit representation
in terms of the Gauss hypergeometric function. There are several parametric
cases when this EGF can be obtained in closed form. One is when the triangle
elements are the generalized Stirling numbers of Hsu and Shiue. Another is when
they are generalized Eulerian numbers of a newly defined kind. These numbers
are related to the Hsu-Shiue ones by an upper binomial transformation, and can
be viewed as coefficients of connection between polynomial bases, in a manner
that generalizes the classical Worpitzky identity. Many identities involving
these generalized Eulerian numbers and related generalized Narayana numbers are
derived, including closed-form evaluations in combinatorially significant
cases.
