Geodesics and curvature in cotangent bundle

Question

My problem concerns formulas given in Tangent and Cotangent Bundles by Yano and Ishihara. In cotangent bundle $T^*M$ we have coordinates: $$(x^a,p_a)=(x^a,x^{\bar{a}})$$ meaning that $\bar{a}=n+a$ where $n=\dim M$. On page 269 they calculate connection coefficients in cotangent bundle and obtain formula: $$\bar{\Gamma}^{\bar{h}}_{ij}=p_a\left(\partial_h\Gamma^{a}_{ji}-\partial_j\Gamma^{a}_{ih}-\partial_i\Gamma^{a}_{jh}+2\Gamma^{a}_{ht}\Gamma^{t}_{ji}\right)$$ where $\Gamma$ is the connection on $M$. Further on geodesics are given: $$\frac{\delta^2p_h}{dt^2}+p_aR_{hji}^{\quad a}\frac{dx^j}{dt}\frac{dx^i}{dt}$$ suggesting that by $R_{hji}^{\quad a}$ authors mean: $$R_{hji}^{\quad a}=\partial_h\Gamma^{a}_{ji}-\partial_j\Gamma^{a}_{ih}-\partial_i\Gamma^{a}_{jh}+2\Gamma^{a}_{ht}\Gamma^{t}_{ji}$$ But usual definition of curvature is: $$R_{hji}^{\quad a}=\partial_j\Gamma^{a}_{ih}-\partial_i\Gamma^{a}_{jh}+\Gamma^{a}_{jt}\Gamma^{t}_{ih}-\Gamma^{a}_{it}\Gamma^{t}_{jh}$$ Finally, I don't known which formula should I use for $R_{hji}^{\quad a}$ in geodesics equation, because these formulas give different curvature coefficients.