As we may know, in the standard approach to the $q$-calculus there are two types of $q$-exponential functions $e_{q}=\sum_{n=0}^{\infty}\frac{z^{n}}{\left[ n\right]_{q}!}$ and $E_{q}(z)=\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n\left(n-1\right) }z^{n}}{\left[ n\right]_{q}!}$. Based on these $q$-exponential functions, also, some new functions are defined whose most of their properties are similar to the exponential function in calculus. Despite having interesting properties similar to the exponential function in calculus, for none of them the above property holds. So, can we define a new $q$-exponential function with similar characteristics to the exponential function in calculus which satisfies the aforesaid property?Any help is appreciated.
Unfortunately, if you are interested in continuous solutions of your functional equation F(x+y)=F(x)F(y), then the exponential functions exp(a x) are the only answer, see the links below for details (note that if F satisfies the above equation, f=ln(F) satisfies the Cauchy functional equation f(x+y)=f(x)+f(y)):
http://math.mit.edu/~stevenj/exponential.pdf
http://math.stackexchange.com/questions/423492/overview-of-basic-facts-about-cauchy-functional-equation
@Prof. Dr. Artur Sergyeyev: Many thanks for your response. What if we generalize the concept of addition? In other words, can we define a new $q$-addition, $+_q$, and then check whether it holds the following identity?
$e_q(x+_qy)=e_q(x)+_qe_q(y).$