Paraphrasing from the authors' introduction: Let \(G=SO(m)\) be a split special orthogonal group of degree \(m=m'+r+1\), \(r\geq 0\), defined over a non-archimedean local field \(k\). Let \(Q\) be a parabolic subgroup of \(G\) whose Levi subgroup is isomorphic to \(SO(m'+1)\times GL(1)^r\), and let \(U\) be the unipotent radical of \(Q\). Let \(K=G(k)\cap GL(m,{\mathcal O})\) and \(K'=G'(k)\cap GL(m',{\mathcal O})\), where \(G'=SO(m')\) is embedded into \(G\) in a particular way and \({\mathcal O}\) denotes the ring of integers of \(k\). Choose a generic character \(\psi:U\rightarrow \mathbb{C}^\times\) which is invariant under the action of \(G'\) on \(U\) and let \(C^\infty(G,\psi)\) denote the space of smooth functions \(f\) satisfying \(\lambda(u)f=\psi(u)f\), where \(\lambda\) denotes the left regular representation. Let \({\mathcal H}={\mathcal H}(G,K)\) and \({\mathcal H}'={\mathcal H}(G',K')\) denote the respective Hecke algebras. Note these act on the space of \(K\times K'\)-fixed vectors \(C^\infty(G,\psi)^{K\times K'}\) by convolution. For given \(\omega\in Hom_{\mathbb{C}-alg}({\mathcal H},\mathbb{C})\) and \(\omega'\in Hom_{\mathbb{C}-alg}({\mathcal H}',\mathbb{C})\), a \textit{Whittaker-Shintani function} attached to \((\omega,\omega')\) is an \((\omega,\omega')\)-eigenvector in \(C^\infty(G,\psi)^{K\times K'}\). This paper shows that the space of such functions is one-dimensional and computes a formula for them in terms of Satake parameters associated to \((\omega,\omega')\). This formula generalizes the well-known formula of Casselman-Shalika, for example.