Generalization of the Genocchi Numbers to their q-analogue

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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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