GENEHATINCt functions tn the theory of numbers. 
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all 1 . . . 
1 a 1 1 . . . 
1 1 a 1 , . . 
1 1 la... 
= (a — 1)" + n (a — 1)" ' 
= (a — 1 (a + n — 1). 
Thence the true generating function 
_ 1 __ 
{1 — a^a\ + {a — 1) (a +1) — {a—Vf {a + 2) %x^x^^ + ... + ( — )"(« —+ n — l)x^x.^...x„}' 
which constitutes a perfect solution of the problem of “ derangement.” 
§ 3. The General Theory Resumed. 
Art. 21. The denominator of a perfect generating function, of the type under 
consideration, is the most general function linear in each of n variables x^, . . . x„. 
Let V„ be the most general linear function of the n quantities, involving 2" — 1 
independent coefficients. 
Art. 22. I enquire, irrespective of arithmetical interpretation or correspondence, 
into the possibility of expressing the fraction 
V 
in a factorized redundant form. 
Art. 23. The coefficients of V„ must be the several coaxial minors of some deter¬ 
minant, and the cjuestion arises : Can a determinant be constructed such that its 
coaxial minors assume given values ? 
The redundant form of order n involves rd coefficients. In general, in order that 
the fraction 
V 
may be expressible in a redundant form, its coefficients must satisfy 
O' 
conditions, and, assuming the satisfaction of these conditions, a redundant form 
involving 
id — (2'' — 1 — o-„) 
arbitrary coefficients can be constructed. 
