I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values.
About the first part of my query, I am not so much looking for an explanation of how to solve the Laplacian but I am looking for an "algorithm" to compute the special functions in terms of which the answers of that come out to be.
There is no dearth of resources for the $Y_{lm}$ on $S^2$! But even for the $Y_{lm}$ I haven't seen a good reference on computing its upper and lower bounds.