An Introduction to Combinatory Analysis

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Since this is an identity we may multiply out the left-hand side and equate the coefficients of the successive powers oi y to zero; obtaining Jh — «i = 0, h^ — ajix + a2 = 0, ^3 — «! 7^2 + a-2 hi — «. J = 0, hn - ajin-i + aiK-i - ... + {-y-an = 0, hn + 2 - CllK-vi + a^hn - ... + {-Tanh = 0, relations which enable us to express any function ^, „ in terms of members of the series ai, a^, ag, ... «».
7. In the applications to combinatory analysis it usually happens that we may regard n as being in
...definitely great and then the relations are simply hi — Oi = 0, ^2 - «! Ai + a. 2 - 0, hs — ajh + a^hi — a-i-O, continued indefinitely.
The before-written identity now becomes (1 — a^y + a^y" - a^y^ + ... Ad inf. ) (1 + hiy + h^y" + h^y^ + ... Ad inf. ) s 1, and herein writing —y for y and transposing the factors we find (1 - hiy + h^y^ - hsy^ + ... Ad inf. ) (1 + a^y + a-^y"^ + a-i'if + ... Ad inf. ) = 1, an identity which is derivable from the former by interchange of the symbols a and h.


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