296 6 Combinatorics and Probability
Solution.Let us analyze the quotient
pk,m(x)=(xk+m− 1 )(xk+m−^1 − 1 )···(xk+^1 − 1 )
(xm− 1 )(xm−^1 − 1 )···(x− 1 ),
which conjecturally is a polynomial with integer coefficients. The main observation
is that
lim
x→ 1pk,m(x)=lim
x→ 1(xk+m− 1 )(xk+m−^1 − 1 )···(xk+^1 − 1 )
(xm− 1 )(xm−^1 − 1 )···(x− 1 )=lim
x→ 1xk+m− 1
x− 1···
xk+^1 − 1
x− 1·
x− 1
xm− 1···
x− 1
x− 1
=
(k+m)(k+m− 1 )···(k+ 1 )
m·(m− 1 )··· 1=
(
k+m
m)
.
With this in mind, we treatpk,m(x)as some kind of binomial coefficient. Recall that one
way of showing that
(n
m)
=m!(nn−!m)!is an integer number is by means of Pascal’s triangle.
We will construct a Pascal’s triangle for the polynomialspk,m(x). The recurrence relation
(
k+m+ 1
m
)
=
(
k+m
m)
+
(
k+m
m− 1)
has the polynomial analogue
(xk+m+^1 − 1 )···(xk+^2 − 1 )
(xm− 1 )···(x− 1 )=
(xk+m− 1 )···(xk+^1 − 1 )
(xm− 1 )···(x− 1 )+xk+^1(xk+m− 1 )···(xk+^2 − 1 )
(xm−^1 − 1 )···(x− 1 ).
Now the conclusion follows by induction onm+k, with the base case the obvious
xk+^1 − 1
x− 1 =x
k+xk− (^1) +···+1.
In quantum physics the variablexis replaced byq=ei, whereis Planck’s con-
stant, and the polynomialspn−m,m(q)are denoted by
(n
m
)
qand called quantum binomial
coefficients (or Gauss polynomials). They arise in the context of the Heisenberg uncer-
tainty principle. Specifically, ifPandQare the linear transformations that describe,
respectively, the time evolution of the momentum and the position of a particle, then
PQ=qQP. The binomial formula for them reads
(Q+P)n=∑nk= 0(
n
k)
qQkPn−k.The recurrence relation we obtained a moment ago,