Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are using Krein's series, but I never heard about this series. The condition is that the zeros of the function must be simple. It resembles the Mittag-Leffler theorem (?). What happens if the zeros are not simple? Is it correct anyway? I see other series like $$\frac{1}{J_0(x)}=\sum _{k=1}^{\infty } -\frac{2 (x-1) (x+1) j_{0,k}}{\left(j_{0,k}-1\right) \left(j_{0,k}+1\right) \left(x-j_{0,k}\right) \left(j_{0,k}+x\right) J_1\left(j_{0,k}\right)}-\frac{-x-1}{2 J_0(1)}-\frac{x-1}{2 J_0(1)}$$ that get it better the above paper.

• Welcome to MO. As it stands, the question is hard to understand, since you keep referring us to the quoted paper. The question should be self-contained, so that the reader can understand it without going to the paper. – Amir Sagiv Dec 27 '19 at 15:52
• It also is not clear, at least to me, what the question here is. Is it a question about the Bessel function? Or a question about Krein's series? You are more likely to get a useful answer if you ask a clear, well-defined question. – Michael Engelhardt Dec 27 '19 at 16:27
• What does "that get it better the above paper" mean? – LSpice Dec 27 '19 at 17:16

I really do not if it class of transformation it is useful but the following converge very quick $$\frac{1}{I_0(x){}^2}=\sum _{k=1}^{\infty } -\frac{2 (x-1) (x+1) \left(-i \left(x^2+1\right) \left(j_{0,k}\right){}^3 I_0\left(i j_{0,k}\right)-i x^2 j_{0,k} I_0\left(i j_{0,k}\right)+2 x^2 I_1\left(i j_{0,k}\right)-i \left(j_{0,k}\right){}^5 I_0\left(i j_{0,k}\right)-2 \left(j_{0,k}\right){}^4 I_1\left(i j_{0,k}\right)\right)}{\left(\left(j_{0,k}\right){}^2+1\right){}^2 \left(\left(j_{0,k}\right){}^2+x^2\right){}^2 I_1\left(i j_{0,k}\right){}^3}-\frac{-x-1}{2 I_0(1){}^2}-\frac{x-1}{2 I_0(1){}^2}$$