(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The problem I am having has to do with part (d) in the picture which I have attached. I have managed to get as far as to determine that the coefficients in the series expansion have the recurrence relation shown below in part (2). From this I think that I have been able to determine that the general form of the coefficients must what is shown in part (3) below. The issue is I am unsure of how to get the proper form of the numerator. Any assistance would be greatly appreciated, thanks!

2. Relevant equations

[itex]a_{n+2}=\frac{n(n+3)-\lambda}{R^{2}(n+2)(n+3)}[/itex] where [itex]a_{o}=1[/itex]

[itex]λ=\frac{2m^{2}}{\omega_{o}^{2}}[/itex] where m is the separation constant

3. The attempt at a solution

[itex]a_{2n}=\frac{something}{(R^{2})^{n}(2n+1)!}[/itex]

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# Problem with numerator in a series expansion

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