The purpose of this thesis is an application of quantum calculus to classical Heat- Burgers’ hierarchy and quantum coherent states. First we construct random walk on q-lattice, corresponding q-heat equation and exact solutions in terms of new family of q-exponential functions. Then we introduce a new type of q-diffusive heat equation and q-viscous Burgers’ equation, their polynomial solutions as generalized Kampe-de Feriet polynomials, corresponding dynamical symmetry and description in terms of Bell polynomials. Shock soliton solutions with fusion and fission of shocks are found and studied for different values of q. The q-semiclassical expansion of these equations in terms of Bernoulli polynomials is derived as corrections in power of ln q. A new class of complex valued function of complex argument as q-analytic functions in terms of q-analytic binomials is introduced and shown that these binomials are generalized analytic functions. As an application, we construct a new type of quantum states as q-analytic coherent states and corresponding q-analytic Fock- Bargmann representation. Then, we extend the concept of q-analytic function for two complex arguments, called double q-analytic functions, which has q-Hermite binomial expansion. As hyperbolic extension, we describe the q-analogue of traveling waves and find the D’Alembert solution of q-wave equation. By introducing q-translation operators we obtain q-binomials, q-analytic and q-anti analytic functions, q-travelling waves and non-commutative binomials. New type of quantum states as Hermite coherent states and Kampe-de Feriet coherent states are studied by generalization of the known Mehler formula. We introduce Golden quantum calculus, and as an application we study Golden quantum oscillator and its angular momentum representations.