scipy.special.wright_bessel#

scipy.special.wright_bessel(a, b, x, out=None) = <wrapped_ufunc 'wright_bessel'>#

Wright’s generalized Bessel function.

Wright’s generalized Bessel function is an entire function and defined as

\[\Phi(a, b; x) = \sum_{k=0}^\infty \frac{x^k}{k! \Gamma(a k + b)}\]

See Also [1].

Parameters:

aarray_like of float: a >= 0
barray_like of float: b >= 0
xarray_like of float: x >= 0
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: Value of the Wright’s generalized Bessel function

Notes

Due to the complexity of the function with its three parameters, only non-negative arguments are implemented.

Added in version 1.7.0.

Array API Standard Support

wright_bessel has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. wright_bessel does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

Digital Library of Mathematical Functions, 10.46. https://dlmf.nist.gov/10.46.E1

Examples

>>> from scipy.special import wright_bessel
>>> a, b, x = 1.5, 1.1, 2.5
>>> wright_bessel(a, b-1, x)
4.5314465939443025

Now, let us verify the relation

\[\Phi(a, b-1; x) = a x \Phi(a, b+a; x) + (b-1) \Phi(a, b; x)\]

>>> a * x * wright_bessel(a, b+a, x) + (b-1) * wright_bessel(a, b, x)
4.5314465939443025