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In studying the eigenvalues of certain tri-diagonal matrices I have encountered a problem of the following form:

{(1+a/x)*2x*sinh[n*arcsinh(x/2)] - 2a*cosh[(n-1)*arcsinh(x/2)]} = 0

where a and n are constants. I'm looking to find n complex roots to this problem, but isolating x is troublesome. I attempted to use the implicit derivatives to obtain an expression for x in terms of a and n but it didn't lead me anywhere.

Is there a general approach to finding the roots of equations of this type? If not, can one find any general properties of the roots, e.g. if they belong to a certain half-plane etc.

The problem may be simplified somewhat if we choose a=-2 and try to find x as a function of n but even here the roots are hard to find.

Any advice would be much appreciated.

Thank you.