On some super-congruences for the coefficients of analytic solutions of certain differential equations

In this paper, we prove some congruences involving the coefficients \(\{A_{n}\}_{n=0,1,2,\ldots }\) ${\{A_n\}}_{n=0,1,2,\dots }$ of the analytic solution \(y_0(z)=\sum _{n=0}^\infty A_nz^n\) $y_0(z)={\sum }_{n=0}^{\infty }{A}_n{z}^n$ of certian differential eqution \({\mathcal {D}}y=0\) $\mathcal{D}y=0$ normalized by the condition \(y_0(0)=A_0=1\) $y_0(0)=A_0=1$ , where \({\mathcal {D}}\) $\mathcal{D}$ is a 4th-order linear differential operator.