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We study the structure of analytic continuation of solutions of an even ranksystem of linear ordinary differential equations of Okubo normal form (ONF). Wedevelop an adjustment of the method by using the Euler integral for evaluatingthe connection formulas of the Gauss hypergeometric function ${}_2F_1(\alpha,\beta, \gamma; x)$ to the system of ONF. We obtain recursive relations betweenconnection coefficients for the system of ONF and ones for the underlyingsystem of half rank.

Recursive calculation of connection formulas for systems of differential equations of Okubo normal form