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We perform global and local analysis of oscillatory and damped sphericallysymmetric fundamental solutions for Helmholtz operators$\big({-}\Delta\pm\beta^2\big)$ in $d$-dimensional, $R$-radius hyperbolic${\mathbf H}_R^d$ and hyperspherical ${\mathbf S}_R^d$ geometry, whichrepresent Riemannian manifolds with positive constant and negative constantsectional curvature respectively. In particular, we compute closed-formexpressions for fundamental solutions of $\big({-}\Delta \pm \beta^2\big)$ on${\mathbf H}_R^d$, $\big({-}\Delta+\beta^2\big)$ on ${\mathbf S}_R^d$, andpresent two candidate fundamental solutions for $\big({-}\Delta-\beta^2\big)$on ${\mathbf S}_R^d$. Flat-space limits, with their corresponding asymptoticrepresentations, are used to restrict proportionality constants for thesefundamental solutions. In order to accomplish this, we summarize and derive newlarge degree asymptotics for associated Legendre and Ferrers functions of thefirst and second kind. Furthermore, we prove that our fundamental solutions onthe hyperboloid are unique due to their decay at infinity. To derive Gegenbauerpolynomial expansions of our fundamental solutions for Helmholtz operators onhyperspheres and hyperboloids, we derive a collection of infinite seriesaddition theorems for Ferrers and associated Legendre functions which aregeneralizations and extensions of the addition theorem for Gegenbauerpolynomials. Using these addition theorems, in geodesic polar coordinates fordimensions greater than or equal to three, we compute Gegenbauer polynomialexpansions for these fundamental solutions, and azimuthal Fourier expansions intwo-dimensions.

Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature