Closed-form Expression for Spherical Integral $I(n, m)$

Question

I am investigating the following integral over the unit sphere $\mathbb{S}^{N-1}$:

$ I(n, m) = \int_{\mathbb{S}^{N-1}} (1 + \langle u, x \rangle)^n (1 + \langle v, x \rangle)^m \, d\omega(x), $

where: $u, v \in \mathbb{S}^{N-1}$ are fixed unit vectors, $n, m \in \mathbb{N}$, $d\omega(x)$ is the standard surface measure on the sphere. Does there exist a $\textbf{closed-form expression}$ for $I(n, m)$ for general exponents $n, m$ in terms of special functions (e.g., hypergeometric functions, Gegenbauer polynomials)?

Answer 1

Without loss of generality we can assume $m\geq n$ and take $v=(1,0,0,\ldots 0)$ and $u=(\cos\phi,\sin\phi,0,0,\ldots 0)$, so $$I(n,m)=\int_{S^{N-1}}(1+x_1\cos\phi+x_2\sin\phi)^n(1+x_1)^m\,dS.$$ I expand $$(1+x_1\cos\phi+x_2\sin\phi)^n(1+x_1)^m=\sum_{r=0}^{n+m}\sum_{s=0}^n C_{r,s}x_1^r x_2^s,$$ $$C_{r,s}=\sum_{j =0}^{\min(r, n - s)} \binom{n}{j + s} \binom{j + s}{j} \binom{m}{r - j} \cos^j\phi \sin^s\phi$$ $$\qquad = \binom{m}{r} \binom{n}{s} \, _2F_1(-r,s-n;m-r+1;\cos \phi)\sin ^s\phi.$$ Only even $r,s$ give a nonzero contribution to the integral over the unit sphere. For even $r,s$ one has the result $$\int_{S^{N-1}}\prod_{i=1}^{N}x_i^{m_i}\,dS=\frac{\prod_{i=1}^N(m_i-1)!!(N-2)!!}{(\sum_{i=1}^N m_i+N-2)!!}$$ $$\Rightarrow \int_{S^{N-1}}x_1^r x_2^s\,dS=\frac{(r-1)!!(s-1)!!(N-2)!!}{(r+s+N-2)!!},$$ hence $$I(n,m)={\sum'}_{r=0}^{n+m}{\sum'}_{s=0}^n C_{r,s}\frac{(r-1)!!(s-1)!!(N-2)!!}{(r+s+N-2)!!},$$ where $\sum'$ means that only even numbers contribute.
There may well be a way to simplify this further.