The Jacobi coefficients $c_{j}^{\ell }\left( \alpha ,\,\beta \right)\,\left( 1\,\le \,j\,\le \,\ell ,\,\alpha ,\,\beta \,>\,-1 \right)$ are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the Jacobi polynomials $P_{k}^{\left( \alpha ,\,\beta \right)}\,\left( k\,\ge \,0,\,\alpha ,\,\beta \,>\,-1 \right)$ into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed, and a direct trace interpretation of the Maclaurin coefficients is presented.

We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ${{\mathbb{R}}^{n}}$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as ${{\left| s \right|}^{p-1}}s$ for $\left| s \right|$ large with $1\,<\,p\,<\,\left( n+5 \right)/\left( n-3 \right)$.

In this paper, a detailed description of the resolvent of the Laplace–Beltrami operator in $n$-dimensional hyperbolic space is given. The resolvent is an integral operator with the kernel (Green’s function) being a solution of a hypergeometric differential equation. Asymptotic analysis of the solution of this equation is carried out.

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra ${\mathcal {O}}$(SLq(N)) are computed.