I am interested in finding analytic expressions for various derivatives of the LegendreP[a,z] w.r.t. a at a = 0. Making use of its hypergeometric representation I was able to find

Derivative[1, 0][LegendreP][0, z] == Log[(1 + z)/2]

The Wolfram functions webpage has a representation for the derivatives in terms of sums. For ν = 0 and m = 1 that formula seems to reduce to

Sum[((-1)^(1 + k) (z)^k BellB[k, -1])/(k^2 Gamma[k]), {k, 1, Infinity}]

which unfortunately does not seem to be resummable.

Overall, my question is:

Is it possible to use Mathematica to quickly obtain analytic expressions for the derivatives of LegendreP[a,z] w.r.t a at a = 0? If yes, how can I do this?

  • $\begingroup$ As it does not yet deserve to be an answer: I am doubtful that Mathematica can do this. Altho there are explicit formulae for low-order derivatives, to get arbitrary derivatives, one has to switch to the ${}_2 F_1$ representation and consider derivatives with respect to the parameters, which can get complicated. $\endgroup$ – J. M.'s technical difficulties Sep 26 '18 at 13:41

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