Reciprocal expansion of modified Bessel function

Question

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.

Answer 1

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