We study the integrals of type $I(a)=\int_{ O_n}\prod u_{ ij}^{ a_{ ij}}\,du$, depending on a matrix $a\in M_{ p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case $a\in M_{ 2\times q}(2\mathbb N)$ to the general case $a\in M_{ p\times q}(\mathbb N)$, (2) the construction of the "best algebraic normalization" of $I(a)$, in the case $a\in M_{ 2\times q}(\mathbb N)$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in M_{ 3\times 3}(\mathbb N)$, (4) a modelling result in the case $a\in M_{ 1\times 2}(\mathbb N)$, in relation with the Euler-Rodrigues formula. Most proofs use various combinatorial techniques.
Categories: math.CO math.PR