2008 - Matt Rogala - Generalization of the Genocchi Numbers to their q-analogue
In the study of functions, it is often useful to derive a more generalized
form of a given function and study it in order to shed new light on the original
function, which is a special case of the object under study. One way in which
to construct such generalizations is through the use of q-series. In this note, we
will discuss some of the tools necessary for constructing these q-analogues of
classical functions, their purpose, and then demonstrate one such construction
on the Genocchi numbers and its close relative, the Euler numbers.
Two methods of generation for the Genocchi numbers will be given, and a
veriﬁcation of the relationship between the Genocchi numbers and the Euler
numbers will be discussed in each case. Following that, the generalization
to a q -analogue of each series will be discussed and the preservation of the
relationship between the two series will be veriﬁed.
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