User:Tohline/Appendix/Mathematics/ToroidalConfusion
Confusion Regarding Whipple Formulae
May, 2018 (J.E.Tohline): I am trying to figure out what the correct relationship is between half-integer degree, associated Legendre functions of the first and second kinds. In order to illustrate my current confusion, here I will restrict my presentation to expressions that give <math>~Q^m_{n - 1 / 2}(\cosh\eta)</math> in terms of <math>~P^n_{m - 1 / 2}(\coth\eta)</math>.
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From equation (34) of H. S. Cohl, J. E. Tohline, A. R. P. Rau, & H. M. Srivastiva (2000, Astronomische Nachrichten, 321, no. 5, 363 - 372) I find:
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<math>~Q^m_{n - 1 / 2}(\cosh\eta)</math> |
<math>~=</math> |
<math>~ \frac{(-1)^n \pi}{\Gamma(n - m + \tfrac{1}{2})} \biggl[ \frac{\pi}{2\sinh\eta} \biggr]^{1 / 2} P^n_{m - 1 / 2}(\coth\eta) \, . </math> |
From Howard Cohl's online overview of toroidal functions, I find:
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<math>~Q^n_{m- 1 / 2}(\cosh\alpha)</math> |
<math>~=</math> |
<math>~(-1)^n ~\Gamma(n-m + \tfrac{1}{2}) \biggl[ \frac{\pi}{2\sinh\alpha} \biggr]^{1 / 2} P^m_{n- 1 / 2}(\coth\alpha)\, , </math> |
Copying the Whipple's formula from §14.19 of DLMF,
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<math>~\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)</math> |
<math>~=</math> |
<math>~ \frac{\Gamma\left(m-n+ \tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2}\right)}\left(\frac{\pi}{2 \sinh\xi}\right)^{1 / 2}P^{n}_{m-\frac{1}{2}}\left(\coth\xi\right) \, . </math> |
As per equation (8) in A. Gil, J. Segura, & N. M. Temme (2000, JCP, 161, 204 - 217), the relationship is:
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<math>~Q_{n-1 / 2}^m (\lambda)</math> |
<math>~=</math> |
<math>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2} \Gamma(n-m+1 / 2)} (x^2-1)^{1 / 4} P_{m-1 / 2}^n(x) \, , </math> |
where, <math>~\lambda \equiv x/\sqrt{x^2-1}</math>. This means, for example, that by setting <math>~x = \coth\eta</math>,
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