2 formula was incorrect

How about re-writing with Cauchy Residue formula?

$$\alpha_n(x) = \frac{(2n+1)!}{2\pi i} \oint \frac{dz}{(z-x)^{2n+2}}\cdot frac{dz}{z^{2n+2}}\cdot \frac{\sinh z }{\cosh z -1 + x}$$

Not sure how it helps you find roots or establish they are real.

1

How about re-writing with Cauchy Residue formula?

$$\alpha_n(x) = \frac{(2n+1)!}{2\pi i} \oint \frac{dz}{(z-x)^{2n+2}}\cdot \frac{\sinh z }{\cosh z -1 + x}$$

Not sure how it helps you find roots or establish they are real.