q-Drivative and h-Derivative. Generalized Taylor’s formula for polynomials Gauss’s binomial formula. q-Binomial coefficients and linear algebra over finite fields. Two Euler’s identities. Jacobi’s triple product identity. q-Hypergeometric functions. Ramanujan product formula. q-Antiderivative, q-Gamma and q-Beta functions. Bernoulli polynomials and Bernoulli numbers. Applications in Number theory and combinatorial analysis. Applications in physics. Applications in statistics and engineering. Non-linear resonance theory of particles.