A zeta function associated to the sub-Laplacian on the unit sphere in \(\mathbb{C}^N\) (Q698310)

The authors consider the zeta function \(\zeta_n=\sum_{p,q\geq 0} m_{p,q} \lambda_{p,q}^{-z}\), associated to the sub-Laplacian on the unit sphere in \(C^N\), where \(\lambda_{p,q}\) are the explicit eigenvalues with multiplicity \(m_{p,q}\). They prove the following formulae: \[ \zeta_{2n+1}(z)=[\sum_{k=1}^n R_{2n+1,2k-1} \zeta(z-2k+1)] [\sum_{k=1}^n R_{2n+1,2k-1} \zeta(z-2k)] \] and \[ \zeta_{2n}(z)=[\sum_{k=1}^{n-1} R_{2n,2k}(2^{z-2k}-1) \zeta(z-2k)] [\sum_{k=1}^n R_{2n,2k}(2^{z-1-2k}-1) \zeta(z-1-2k)], \] where \(\zeta\) is the classical Riemann zeta function and the coefficients \(R_{n,k}\) are explicitly given.