The first q-functions

We introduce the first q-functions, the tilde operator, two other tilde operators and the △-operator. The q-integral $$ \int_{0}^{a} f(t,q) \,d_q(t)\equiv{a}(1-q) \sum_{n=0}^{\infty}f(aq^{n},q)q^{n} $$ can be written in the form $$ \int f d\mu=\sum_{n=0}^{\infty}b_n\mu(E_n), $$ where b n means the function value f(aq n ,q) times a and μ(E n )=(1−q)q n denotes the measure in the point x=aq n . We use a σ-algebra , where the sets $\{E_{n}\}_{0}^{\infty}$ are disjoint. The q-binomial coefficients are an important tool for the various operator formulas, which are mostly proved by induction. Several Gould and Carlitz q-binomial coefficient identities are given together with Cigler’s equivalent operational method. We present five q-exponential functions, limits, inequalities, and the corresponding q-trigonometric functions. The q-additions are the perfect partners to produce addition formulas for these functions. We give a complex analysis description of elliptic functions and we discuss Theta functions and their connection to the Γ q function via the q-analogue of the Euler reflection formula. We show some graphs that illustrate this formula. Finally, we mention that the parameter augmentation is the inverse of limits a→±∞.