Theorem 3.1 Given the multinomial anchor
we define its set of (p,q)-satellites to consist
at level (n+p) of the m coefficients
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at level (n+q) of the m coefficients
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and at level (n+p+q) of the m(m-1) coefficients
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(e, h) = (p,q)
or (q,p).
Then these m(m+1) coefficients may be partitioned into m sets,
each containing (m+1) coefficients, in such a way that the product of
the coefficients in each set is the number N(p,q) given by
N(p,q) = |
(n+p)!(n+q)!((n+p+q)!)m-1
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Õ |
(ki+p)! |
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(ki+q)! ( |
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ki!)m-1 |
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, (3.2) |
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Corollary 3.2 The number N(p,q) is an integer.
For N(p,q) is a rational number whose mth
power is an integer.
Note that p, q may take any integer values such that the factorial
functions in (3.2) are applied to non-negative integers.
Of course the partitioning function is exactly that used in [6]; theirs is the hard work!
We may, just as for binomial coefficients, generalize the content of
Section 1
substantially beyond the domain of multinomial coefficients. Indeed, in
Section 1, all we require of the basic function of
n, k1, k2,...,km
is that it be a separable
function; thus we may replace the multinomial coefficient by a function
F(n,k1,k2,...,km)
= f(n)f1(k1)...fm(km) |
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, (3.3) |
An example of such a separable function is the q-analogue of
the multinomial coefficient. This is the Gaussian polynomial (see [7]),
obtainable from the multinomial coefficient
by replacing each
occurrence of r! by
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(qr-1)(qr-1-1)...(q-1)
(q-1)r |
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of course, 0!q is just 1. The Gaussian polynomial
may be regarded as the coefficient of
a1k1a2k2...amkm
in the expansion of
(a1+a2+...+am)n
in the algebra over R generated by
(a1,a2,...,am; q)
subject to
ajai=qaiaj, i<j,
qai=aiq. Notice that, in this
generalization of the material of Section 1, it is not necessary that the
functions f1, f2,...,fm
be the same; however, if F in (3.3) is (as in the special case of the multinomial
coefficients and in the example of the Gaussian polynomials) invariant under
the action of the symmetric group Sm, then the functions
f1, f2,...,fm
may be chosen to be the same, given that F is
separable.
The material of Section 2 makes it obvious how we would extend the
definition of F, given by (3.3) to the domain of arbitrary
integers n, k1, k2,...,km.
The formula (2.5) is easily generalized by replacing the multinomial coefficient by
F; indeed, the generalization does not even require that F be
separable. However, one has to be careful in assigning a value to F
in the regions in which the multinomial coefficient takes the value 0;
this problem was first encountered in extending the harmonic triangle in
[4]. We do not want to put any stress on this technical point
here, since it is in the regions where formula (2.5) holds that the
real interest lies.
Finally, we may obviously generalize Theorem 3.1, provided that we insist in
(3.3) that F be invariant under the action of Sm. We leave
the details to the reader.