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Herz (1955) provides the following equality: $$ A_\delta(-\lambda) - A_\delta(-\lambda)\lambda^{-\delta} = -\sin(\pi\delta)B_\delta(\lambda)/\pi $$ where $A_\delta$ and $B_\delta$ are the Bessel functions of matrix argument of first type and second type, in the case when the matrices are $1\times 1$.

However the left member is zero for $\lambda=1$, but not the right member. Is there something wrong in this equality? What is the error?

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  • $\begingroup$ Since it is for $1\times1$ matrices, that is, scalars, couldn't each term ($A_\delta$, $B_\delta$) be written without Bessel functions of matrix arguments? $\endgroup$ Commented Sep 18, 2023 at 12:22
  • $\begingroup$ @username No idea. In the paper they are defined for matrices only (if I correctly remember, not sure... but almost sure). But would it change something? $\endgroup$ Commented Sep 18, 2023 at 12:40
  • $\begingroup$ Well you say that this identity holds for 1x1 "matrices". But in that case, the complicated notations for Bessel Functions of matrix argument simplify greatly since tr(X)=X, |k|=k etc...all in all it becomes normal Gamma functions and normal factorials. With any luck, if the notations are consistent, $A$ will become $J$ and $B$ will become $Y$ (or another natural pair of non matrix argument Bessel functions) and then you can just look up what the corresponding identity is in the DLMF. $\endgroup$ Commented Sep 21, 2023 at 19:19
  • $\begingroup$ @username I didn't know (or didn't remember) that $A$ -> $J$ and $B$ -> $Y$. Thanks. Will take a look at DLMF. $\endgroup$ Commented Sep 22, 2023 at 7:17

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