Fix $c \in \mathbb R$ and an integer $k \ge 0$, and consider the function $h:\mathbb R \to \mathbb R$ defined by $h(x) := (x+c)^k$. Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$, and consider the function $u:[-1,1] \to \mathbb R$ defined by
$$ u(t) := \mathbb E[h(X_1)h(tX_1+(1-t^2)^{1/2} X_2)]. $$
Question. Is there a closed-form formula (perhaps in terms of special functions) which gives the value of $u(t)$ for all $t \in [-1,1]$ ?
Observation. Let $Z \sim N(0,1)$. Becuase the Wasserstein distance between $X_1$ and $Z$ is of order $\mathcal O(1/d)$ (see this post), in the limit $d \to \infty$, we have $$ u(0) = \mathbb E[h(X_1)^2] = \mathbb E[(X_1+c)^{2k}] \to \mathbb E[(Z+c)^{2k}], $$ and we recognize the rightmost term as the $2k$th moment of $N(c,1)$.