I know that if $f(x)=g(\frac{x}{1-x})$, then $f(\frac{x}{1+x})=g(x)$ when $f(x)$ and $g(x)$ are function.
When I read about generating functions , I saw that "Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. Generating functions are sometimes called generating series" in wiki.
Now assume that we have two generating function such that $W(x,y) = S (\frac{x}{1-x},\frac{y}{1-y})$ where $W= \frac{1}{1-(x+y)}$ and $S$ represent the smirnov words. Then the book says that $W(\frac{x}{1+x},\frac{y}{1+y})=S(x,y)$.
My question is that if generating functions are not actually function, then how do they satify the property of functions. Can you prove the foregoing relation about the generating functions ?
Generating functions, are, in general, formal power series with coefficients from $\mathbb{R}$. Formal power series form a ring: you can add them component-wise, you can multiply them by constants, you can multiply them by other formal power series, since coefficient before each term of the resulting series is a finite sum: $$a(z)b(z)=\left(\sum_{k=0}^{\infty}a_k\cdot z^{k}\right) \left(\sum_{l=0}^{\infty}b_{l} \cdot z^{l}\right)=\sum_{n=0}^{\infty}z^{n}\sum_{k=0}^{n}a_{k}b_{n-k}$$
The functional composition of formal power series is well-defined as well, if in the expression $A(B)$ where $A, B$ are formal power series, $B$ has no term of degree 0 (otherwise in the resulting series coefficients are not finite sums anymore): $$a(b(z))=\sum_{k=0}^{\infty}a_k\left(\sum_{l=1}^{\infty}b_{l}\cdot z^{l} \right)^{k}$$
The notation $W\left(\frac{x}{1-x},\frac{y}{1-y}\right)=S(x,y)$ means the formal power series $W(...)$ and $S(x,y)$ coincide in each term, and $$W\left(\frac{x}{1-x},\frac{y}{1-y}\right)$$ should be understood as composition of formal power series: $$\frac{x}{1-x}=x\sum_{k=0}^{\infty}x^{k}=\sum_{k=1}^{\infty}x^{k}$$ (note that the term of degree 0 is not present).
You can also work with formal power series of several variables. Again, they are treated as ordered sets of coefficients, but these sets have several indices (power of $x$, power of $y$, power of $z$ etc.). Substitution, multiplication and summation are again defined component-wise. For instance, the formal power series defined by the function of two variables $1/(1-(x+y))$ is $$\frac{1}{1-(x+y)}=\sum_{n=0}^{\infty}(x+y)^{n}=\sum_{n=0}^{\infty}\sum_{k=0}^{n}{n \choose k}x^{k}y^{n-k}$$ Since $k+(n-k)=n$, the $n$-th "small" sum is homogeneous of total degree $n$, and you can't simplify the expression any further (however, you need to choose some convention by which you sort coefficients inside such series).
If two formal power series have intersecting domains of convergence, and define some functions on the intersection, the algebraic relations between these formal power series are the same as between their corresponding functions in the region of intersection.