We will argue in this section that the rule for defining trinomial
coefficients
where n is a negative integer, should be as follows:
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s ³ 0, when |
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= (-1)n-t |
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when |
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= (-1)n-s |
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when |
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= (-1)n-r |
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(2.1) |
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We base ourselves on the observation that, if (a+b+c)n
is to be represented by a convergent power series, then there are only 3
possibilities (compare [3, 4] and
(2.6) below). We may
have
| (a+b+c)n
= |
å
r,s ³ 0 |
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arbsct
(2.2) |
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or
| (a+b+c)n
= |
å
r,t ³ 0 |
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arbsct
(2.3) |
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or
| (a+b+c)n
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å
s,t ³ 0 |
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arbsct
(2.4) |
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where the coefficients
are given by the appropriate formulae in (2.1). We will be content to show that the
formula (2.2) yields a convergent power series in the open subset of
R3 given by
|a+b|<c, |a|<|c|, |b|<|c|.
For, if |a+b|<c,
then, by the rule given in [3, 4],
| (a+b+c)n
= |
å
u ³ 0,
u+t
= n |
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(a+b)uct,
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where
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= (-1)u |
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. |
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Now, since |a| < |c|, |b| < |c|, we may expand
(a+b)u by the binomial theorem and obtain
a convergent double power series
| (a+b+c)n
= |
å
r,s ³ 0,
r+s = u, u+t = n |
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arbsct, |
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and
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= (-1)n-t |
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. |
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Thus we obtain formula (2.2).
Obviously the rules (2.1) extend to m-multinomial coefficients.
Thus, without going into details, we obtain the rule:
Let n be a negative integer. Then, with
unless exactly one of k1, k2,..., km is
negative. If ki is negative, then
It is interesting to remark that (2.5) generalizes the rule for
defining
in the blade
n < 0,
r ³ 0, s
< 0
of the Pascal Hexagon, or Windmill (see [3, 4], namely;
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= (-1)r |
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. (2.6) |
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However, to construct the generalization we must replace (-1)r in
(2.6) by the equivalent (-1)n-s.
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