Symmetric Quantum Calculus

The q- and h-differentials may be “symmetrized“ in the following way, (26.1)$$ \tilde d_q f(x) = f(qx) - f(q^{ - 1} x), $$(26.2)$$ \tilde d_h g(x) = g(x + h) - g(x - h), $$ where as usual, q ≠ 1 and h ≠ 0. The definitions of the corresponding derivatives follow obviously: (26.3)$$ \tilde D_q f(x) = \frac{{\tilde d_q f(x)}} {{\tilde d_q x}} = \frac{{f(qx) - f(q^{ - 1} x)}} {{(q - a^{ - 1} )x}}, $$(26.4)$$ \tilde d_h g(x) = \frac{{\tilde d_h g(x)}} {{\tilde d_h x}} = \frac{{g(x + h) - g(x - h)}} {{2h}}. $$ 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.