Norm of product

sixth order identity

Sixth and higher order terms produce other series identities.

The non trivial identities for real numbers obtained to sixth order series expansion of the product identity $\prod _{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=\prod _{i=1}^{n}\left(1-a_{i}^{2}\right)\prod _{i=1}^{n}\left(1-b_{i}^{2}\right)$ are

$\sum _{i<j}^{n}\left[a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)\right]+\sum _{i<j<k}^{n}\left[a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right]=\left(\sum _{i=1}^{n}b_{i}^{2}\right)\left(\sum _{i<j}^{n}a_{i}^{2}a_{j}^{2}\right)$

and its counterpart, obtained by interchanging the variables $a$ and $b$.

Expand the product identity in series up to sixth order. The LHS is $\prod _{i=1}^{n}\left(1-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)=1+\sum _{i=1}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)+\sum _{i<j}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)\left(-a_{j}^{2}-b_{j}^{2}+a_{j}^{2}b_{j}^{2}\right)+\sum _{i<j<k}^{n}\left(-a_{i}^{2}-b_{i}^{2}+a_{i}^{2}b_{i}^{2}\right)\left(-a_{j}^{2}-b_{j}^{2}+a_{j}^{2}b_{j}^{2}\right)\left(-a_{k}^{2}-b_{k}^{2}+a_{k}^{2}b_{k}^{2}\right)+{\mathcal {O}}^{7+}.$ Consider only the sixth order terms ${\mathcal {O}}^{6}\left({\textrm {LHS}}\right)=-\sum _{i<j}^{n}\left[a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)+b_{i}^{2}b_{j}^{2}\left(a_{i}^{2}+a_{j}^{2}\right)\right]-\sum _{i<j<k}^{n}a_{i}^{2}a_{j}^{2}a_{k}^{2}+b_{i}^{2}b_{j}^{2}b_{k}^{2}-\sum _{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)-\sum _{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)$ The RHS of the product identity is similarly expanded in series up to sixth order

$\prod _{i=1}^{n}\left(1-a_{i}^{2}\right)\prod _{i=1}^{n}\left(1-b_{i}^{2}\right)=\left(1-\sum _{i=1}^{n}a_{i}^{2}+\sum _{i<j}^{n}a_{i}^{2}a_{j}^{2}-\sum _{i<j<k}^{n}a_{i}^{2}a_{j}^{2}a_{k}^{2}+{\mathcal {O}}^{7+}\right)\left(1-\sum _{i=1}^{n}b_{i}^{2}+\sum _{i<j}^{n}b_{i}^{2}b_{j}^{2}-\sum _{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}+{\mathcal {O}}^{7+}\right),$ and only sixth order terms retained ${\mathcal {O}}^{6}\left({\textrm {RHS}}\right)=-\sum _{i<j<k}^{n}a_{i}^{2}a_{j}^{2}a_{k}^{2}-\sum _{i<j<k}^{n}b_{i}^{2}b_{j}^{2}b_{k}^{2}-\left(\sum _{i=1}^{n}a_{i}^{2}\right)\left(\sum _{i<j}^{n}b_{i}^{2}b_{j}^{2}\right)-\left(\sum _{i=1}^{n}b_{i}^{2}\right)\left(\sum _{i<j}^{n}a_{i}^{2}a_{j}^{2}\right).$ These two results are equated for equal powers of $a^{n}b^{m}$. The terms $a^{6}$ and $b^{6}$ give trivial identities whereas the terms involving $a^{4}b^{2}$ and $a^{2}b^{4}$ give the non trivial sixth order identities

$\sum _{i<j}^{n}a_{i}^{2}a_{j}^{2}\left(b_{i}^{2}+b_{j}^{2}\right)+\sum _{i<j<k}^{n}\left(a_{i}^{2}a_{j}^{2}b_{k}^{2}+a_{i}^{2}b_{j}^{2}a_{k}^{2}+b_{i}^{2}a_{j}^{2}a_{k}^{2}\right)=\left(\sum _{i=1}^{n}b_{i}^{2}\right)\left(\sum _{i<j}^{n}a_{i}^{2}a_{j}^{2}\right)$

$\sum _{i<j}^{n}b_{i}^{2}b_{j}^{2}\left(a_{i}^{2}+a_{j}^{2}\right)+\sum _{i<j<k}^{n}\left(a_{i}^{2}b_{j}^{2}b_{k}^{2}+b_{i}^{2}a_{j}^{2}b_{k}^{2}+b_{i}^{2}b_{j}^{2}a_{k}^{2}\right)=\left(\sum _{i=1}^{n}a_{i}^{2}\right)\left(\sum _{i<j}^{n}b_{i}^{2}b_{j}^{2}\right).$