Let $D(a)$ denote the stated constant term and suppose every $a_i>0$. The rational identity
$$
\sum_{k=1}^n
\prod_{j\ne k}\left(1-\frac{X_k}{X_j}\right)^{-1}=1
$$
can be verified after clearing denominators, or by <Lagrange interpolation polynomial>[Lagrange interpolation]. Multiplying it by the Dyson product and taking constant terms gives the recursion
$$
D(a_1,\ldots,a_n)=
\sum_{k=1}^nD(a_1,\ldots,a_k-1,\ldots,a_n).
$$
The <multinomial coefficient>
$$
M(a)=\frac{(a_1+\cdots+a_n)!}{a_1!\cdots a_n!}
$$
obeys the same recursion by the multinomial form of <Pascal's identity>.

If $a_k=0$, taking the constant term in $X_k$ forces the zero term from every factor involving $X_k$ and reduces the expression to the $(n-1)$-variable Dyson product with $a_k$ omitted. The same boundary reduction holds for $M(a)$. Finally $D(0,\ldots,0)=M(0,\ldots,0)=1$. Induction on $n$ and on $a_1+\cdots+a_n$ therefore proves the <Dyson constant-term identity>
$$
\boxed{D(a)=M(a)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 109 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-109/3/a/solution