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I encountered a problem copying and pasting your code, as HypergeometricU and LaguerreL begin on new lines, but I believe you meant for them to be part of the definitions for DI1 and DI2. Fixing that, the problem of non-numerical integrand comes from undefined global U, which can be set inside Block, and FindRoot attempting to evaluate its argument ...

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I tried to solve this problem by explicitly adding the order of terms you want to drop, and simplifying the result, but ended up being more confused by the behavior of FullSimplify. Nevertheless, Factor[Series[BesselI[n, z], {z, Infinity, 1}] + O[z, Infinity]^(1/2) Exp[-z]] works, and you can get rid of O[_] with Normal if desired. Factor is needed to ...

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expr = -((2 23 (-1 + k^2 JacobiSN[h Ω, k]^4))/ (k^2 JacobiCN[h Ω, k] JacobiDN[h Ω, k] JacobiSN[h Ω, k]^2)); rules = {_JacobiSN -> sn, _JacobiCN -> cn, _JacobiDN -> dn}; expr /. rules

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Rules are your friends. (And I should use them more myself.) Here's a conditional rule that should help: gamRule = {Gamma[x_] /; x > 1 -> (x - 1) Gamma[x - 1]}; (111 Gamma[5/4]^3)/(-96 Gamma[9/4]^3 + 40 Gamma[5/4]^2 Gamma[13/4]) //. gamRule (* -(37/25) *) In this particular example, FullSimplify is not needed but the use of //. (ReplaceRepeated) ...

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I realized I did not answer what the OP was asking in my earlier answer. That was the reason for the misbehaviour, here's the solution to the problem: The solution can be obtained in a piecewise format: the coefficient at $(x-1)^l$ is \alpha_l = \begin{cases} a_l + b_l & l ≤ n+2\ \mbox{and even}, \\ a_l + c_l & l ≤ n+2\ \mbox{and odd}, \\ a_l &...

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It's a problem of genericity. The output of the command Series[ Hypergeometric2F1[-1 - n/2, -n, -n - 2, (1 - z)], {z, 0, 0}] (prior to substituting for $n$) is valid for almost all real values of $n$ but fails for those which are integer. The reason is that the hypergeometric function changes behaviour in these: hg = Hypergeometric2F1[-1 - n/2, -n, -n - 2, (...

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