Error bounds for a uniform asymptotic expansion of the Legendre function 𝑃_{𝑛}^{-π‘š}(π‘π‘œπ‘ β„Žπ‘§)

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Abstract
For fixed m m with m + 1 2 > 0 m + \frac {1}{2} > 0 , an asymptotic expansion for large n n is derived for the Legendre function P n βˆ’ m ( cosh ⁑ z ) P_n^{ - m}\left ( {\cosh z} \right ) ,which is uniformly valid for z z in the unbounded interval [ 0 , ∞ ) \left [ {0, \infty } \right ) . Our method is based on an integral representation of this function. The coefficients in the expansion satisfy a recurrence relation. Simple computable bounds are also constructed for the error terms associated with the expansion.

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