scipy.special.lpmv

scipy.special.lpmv(m, v, x, out=None) = <ufunc 'lpmv'>

Associated Legendre function of integer order and real degree.

Defined as

\[P_v^m = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_v(x)\]

where

\[P_v = \sum_{k = 0}^\infty \frac{(-v)_k (v + 1)_k}{(k!)^2} \left(\frac{1 - x}{2}\right)^k\]

is the Legendre function of the first kind. Here \((\cdot)_k\) is the Pochhammer symbol; see poch.

Parameters:

marray_like

Order (int or float). If passed a float not equal to an integer the function returns NaN.

varray_like

Degree (float).

xarray_like

Argument (float). Must have |x| <= 1.

outndarray, optional

Optional output array for the function results

Returns:

pmvscalar or ndarray

Value of the associated Legendre function.

Notes

Note that this implementation includes the Condon-Shortley phase.

lpmv has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	⚠️ no JIT	⛔
Dask	✅	n/a

See Support for the array API standard for more information.

References

[1]

Zhang, Jin, “Computation of Special Functions”, John Wiley and Sons, Inc, 1996.