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Is there any known formulae for the derivative of the Bessel function with respect to the order of the Bessel function?

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  • $\begingroup$ which Bessel functions are you talking about? There are lots of different kinds. $\endgroup$ – Scott Morrison Nov 4 '09 at 0:26
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series... from Maple

$${\frac {d}{dr}}{{\rm J}_r\left(2\right)}= \sum _{k=0}^{\infty }{\frac { \left( -1 \right) ^{k+1}\Psi \left( 1+r+k \right) }{\Gamma \left( 1+r+k \right) \Gamma \left( 1+k \right) }}$$

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I suggest looking at Landau's paper:http://www.emis.de/journals/EJDE/conf-proc/04/l1/landau.pdf

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Abramowitz and Stegun give a couple of special cases but don't give a general result. Starting from some of the integral or series representations and differentiating you can get a corresponding integral or series for the derivative, but I would guess that it's unlikely to simplify to a "known" function in the general case. An example they give is (for the spherical Bessel function $j_\nu(x)$):

$$[ \frac{d}{d\nu} j_\nu(x) ]_{\nu=0} = \frac{\pi}{2x}(\operatorname{Ci}(2x)\sin x - \operatorname{Si}(2x)\cos x)$$

They also give examples evaluated at $\nu=-1$ and similar results for the case of the "other" spherical bessel $y_\nu(x)$.

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As I understand it you are looking for $D_\nu(x):=\frac{d}{d\nu} J_\nu(x)$. Perhaps you would like to explain a bit why you are looking at $D_\nu$?

I have worked a bit with the Legendre functions of the first ($P_\nu$) and second kind ($Q_\nu$). Where my primary interest was to find estimates in $\nu$ and both parameters. To find such estimates basically I used relations together with integral representations. At one point I estimated an integral expression for $\frac{d}{d\nu} Q_\nu(x)$ in order to see that for fixed $x>1$ it is decreasing with respect to $\nu$. (The main reason to these studies was to prove a Tauberian theorem for spaces like $L^1_w(G//K)$ - the double coset space of $G=SL(2,R)$ - where $\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx$ is the Fourier transform.)

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The following paper might also be interesting:

http://www.tandfonline.com/doi/full/10.1080/10652469.2016.1164156

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