# What is the correct formula for $n$-dimensional spherical harmonics?

In the Wikipedia article, the formula for $$n$$-dimensional spherical harmonics is given as $$Y_{\ell_1, ..., \ell_{n-1}}( \theta_1, \dots \theta_{n-1} ) = \frac{1}{\sqrt{2\pi}} e^{i \ell_1 \theta_1} \prod_{j = 2}^{n-1} {}_j \bar{P}^{\ell_{j-1}}_{\ell_j}( \theta_j ) ,$$ where the indices satisfy $$| \ell_1 | \le \ell_2 \le ... \le \ell_{n−1}$$ and the eigenvalue is $$− \ell_{n-1} ( \ell_{n-1} + n − 2)$$, i.e., $$\Delta_{S^{n-1}} Y_{\ell_1, ..., \ell_{n-1}}( \theta_1, \dots \theta_{n-1} ) = - \ell_{n-1} ( \ell_{n-1} + n − 2) Y_{\ell_1, ..., \ell_{n-1}}( \theta_1, \dots \theta_{n-1} ) ,$$ where $$\Delta_{S^{n-1}}$$ is the Laplace-Beltrami operator on the $$n$$-sphere $$S^{n-1}$$. The functions $${}_j \bar{P}^{\ell_{j-1}}_{\ell_j}( \theta_j )$$ in the product are defined in terms of the Legendre function: $${}_j \bar{P}^\ell_{L}( \theta ) = \sqrt{\frac{2L + j - 1}{2} \frac{(L + \ell + j - 2)!}{(L - \ell)!}} \sin^{\frac{2 - j}{2}}( \theta ) P^{-\left(\ell + \frac{j-2}{2}\right)}_{L+\frac{j-2}{2}}(\cos \theta) ,$$ where $$P^{-\mu}_{\gamma}( x ) = \frac{1}{\Gamma( 1 + \mu )} \left( \frac{1 - x}{1 + x} \right)^{\mu / 2} F\left( - \gamma , \gamma + 1; 1 + \mu; \frac{1 - x}{2} \right) ,$$ and $$F\left( \alpha , \beta; \gamma; z \right)$$ is the hypergeometric function. However, in page 1554 in Higuchi (1987), a paper also cited in the Wikipedia article, the formula is given as $$Y_{\ell_1, ..., \ell_{n-1}}( \theta_1, \dots \theta_{n-1} ) = \frac{1}{\sqrt{2\pi}} e^{i \ell_1 \theta_1} \prod_{j = 2}^{n-1} {}_j \bar{P}^{\ell_{n-2}}_{\ell_j}( \theta_j ) ,$$ where the difference is in the superscript of $$\bar{P}$$ and all other quantities are same.

What is the correct formula for the $$n$$-dimensional spherical harmonics?