Quantum calculus

Abstract: This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontin- uous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational itera- tion method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effec- tive analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional com- plex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.

Abstract: 1 Introduction.- 2 The different languages of q.- 3 Pre q-Analysis.- 4 The q-umbral calculus and the semigroups. The Norlund calculus of finite diff.- 5 q-Stirling numbers.- 6 The first q-functions.- 7 An umbral method for q-hypergeometric series.- 8 Applications of the umbral calculus.- 9 Ciglerian q-Laguerre polynomials.- 10 q-Jacobi polynomials.- 11 q-Legendre polynomials and Carlitz-AlSalam polynomials.- 12 q-functions of many variables.- 13 Linear partial q-difference equations.- 14 q-Calculus and physics.- 15 Appendix: Other philosophies.

Abstract: In this paper we initiate the study of quantum calculus on finite intervals. We define the qk-derivative and qk-integral of a function and prove their basic properties. As an application, we prove existence and uniqueness results for initial value problems for first- and second-order impulsive qk-difference equations. MSC: 26A33; 39A13; 34A37

Abstract: The q- and h-differentials may be “symmetrized“ in the following way, $$ \tilde d_q f(x) = f(qx) - f(q^{ - 1} x), $$ (26.1) $$ \tilde d_h g(x) = g(x + h) - g(x - h), $$ (26.2) where as usual, q ≠ 1 and h ≠ 0. The definitions of the corresponding derivatives follow obviously: $$ \tilde D_q f(x) = \frac{{\tilde d_q f(x)}} {{\tilde d_q x}} = \frac{{f(qx) - f(q^{ - 1} x)}} {{(q - a^{ - 1} )x}}, $$ (26.3) $$ \tilde d_h g(x) = \frac{{\tilde d_h g(x)}} {{\tilde d_h x}} = \frac{{g(x + h) - g(x - h)}} {{2h}}. $$ (26.4) We are going to concern ourselves briefly with symmetric q-calculus only, since it is important for the theory of some algebraic objects called quantum groups.