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We compute power-suppressed corrections to the \eta\gamma and\eta^{\prime}\gamma transition form factorsQ^2F_{\eta(\eta^{\prime})\gamma}(Q^2) arising from the end point regions x \to0,1 by employing the infrared-renormalon approach. The contribution to the formfactors from the quark and gluon content of the \eta,\eta^{\prime} mesons istaken into account using for the \eta-\eta^{\prime} mixing the SU_f(3) singlet\eta_1 and octet \eta_8 basis. The theoretical predictions obtained this wayare compared with the corresponding CLEO data and restrictions on the inputparameters (Gegenbauer coefficients) B_2^q(\eta_1), B_2^g(\eta_1), andB_2^q(\eta_8) in the distribution amplitudes for the \eta_1,\eta_8 states withone nonasymptotic term are deduced. Comparison is made with the results fromQCD perturbation theory.Comment: 25 pages, RevTeX4 used. 9 figures as EPS files. Text significantly  changed to include variation of theoretical parameters. Figures modified.  Corrected typo in equation (34) and trivial mistake in $\beta_1$-coefficient.  References added. Conclusions unchange

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

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