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I am reading this paper. 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.

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.

I read the paper https://iopscience.iop.org/article/10.1088/1742-6596/937/1/012047/pdf using krein,s series the autor say but I never heard about this series the conditions is that the zeros of the function must be simples ,it resembles of mittag leffler theorem ?? what happen if the zeros are not simples it is coorect 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

I am reading this paper. 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.