Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/20

6 Thence we obtain

Since this is an identity we may multiply out the left-hand side and equate the coefficients of the successive powers of $$y$$ to zero; obtaining

relations which enable us to express any function $$h_w$$ in terms of members of the series $$a_1$$, $$a_2$$, $$a_3$$,… $$a_n$$.

7. In the applications to combinatory analysis it usually happens that we may regard $$n$$ as being indefinitely great and then the relations are simply continued indefinitely.

The before-written identity now becomes

and herein writing $$-y$$ for $$y$$ and transposing the factors we find

an identity which is derivable from the former by interchange of the symbols $$a$$ and $$h$$.

There is thus perfect symmetry between the symbols and it follows as a matter of course that in any relation connecting the quantities $$a_1$$, $$a_2$$, $$a_3$$,… with the quantities $$h_1$$, $$h_2$$, $$h_3$$,… we are at liberty to interchange the symbols $$a$$, $$h$$. This interesting fact can be at once verified in the case of the relations $$h_1-a_1 = 0$$, etc.

Solving these equations we find