How can I deal with the derivatives bessel with Respect to Order?

[Jiaqi-knight]

Sir, Thankyou for your good job.
I want to ask how can I deal with the derivatives bessel with Respect to Order?
Is this package can help me or not.

[simonbyrne]

I'm not sure what you mean. Can you expand on what you want, or provide a link?

[stevengj]

I think you mean things like ∂/∂ₐ Jₐ(z). The answer is no, we don't have anything like this right now — as far as I know you can't implement it without implementing an entirely new special function.

My student @rpestourie experimented with this for a while, and got something working for Jₐ(z) via a combination of the power series for small z and the asymptotic series for large z, but the case of Yₐ(z) was a lot trickier because of the divergences leading to huge cancellation errors at some points. You can see his code for ∂/∂ₐ Jₐ(z) here: https://nbviewer.jupyter.org/gist/stevengj/bc113b9d84b8bf26560e6749781f128c

[stevengj]

(You can also do compute it by quadrature, e.g. with quadgk, using the integral formulation, as in Ĵ_quad(n,z) in the notebook I linked, but that is a lot slower and we only used it for validation.)

[devmotion]

Copied from JuliaDiff/ChainRules.jl#208:

In tandfonline.com/doi/pdf/10.1080/10652469.2016.1164156, closed-form expressions were derived for the derivatives of Bessel functions with respect to the order.

These derivatives are useful, e.g., when working with the Matern kernel (see JuliaGaussianProcesses/KernelFunctions.jl#116 (comment)).

It feels a bit problematic that these expressions involve hypergeometric functions and hence probably would introduce a dependency on HypergeometricFunctions.jl.

I noticed that there was some discussion about HypergeometricFunctions in #175, not sure how related it is.