Definition 5

  The W-functions W₁(x) and W₂(x) will be said to belong to the same type if, for some constants a > 0 and b, the following equations hold:

$$\begin{array} {r@{~}c@{~}l} W_2(x) & = & W_1 (ax+b), \\ W_1 ... ...splaystyle W_2 \left( \frac{x}a - \frac{b}a \right).\end{array}$$

Since the property of belonging to the same type is symmetrical and transitive, the totality of all W-functions fall into mutually disjoint types.