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### Description

The $\eta^\prime$-meson energy spectrum in the inclusive $\Upsilon (1S) \to \eta^\prime g g g \to \eta^\prime X$ decay measured by the CLEO Collaboration in 2002 allowed one to constrain the lowest Gegenbauer coefficients $B^q_2$ and $B^g_2$ of the quark-antiquark and gluonic distribution amplitudes of the $\eta^\prime$-meson entering the $\eta^\prime g^* g$ effective vertex function (EVF). The fitting procedure of the CLEO data on the hard part of the $\eta^\prime$-meson energy spectrum was based on the theoretical expression calculated in the leading-order perturbative QCD in the static-quark limit for the orthoquarkonium. The resulting constraints were combined with the existing ones on these coefficients from an analysis of the $\eta^\prime - \gamma$ transition form factor. The updated measurements of the $\eta^\prime$-meson energy spectrum by the CLEO Collaboration in 2006 results a worser consistency with theoretical expectations and require some improvements from a theory. As a first step, we assume the dependence of the strong coupling constant in the $\eta^\prime g^* g$ EVF on the quark energy the $\eta^\prime$-meson and repeat the fitting procedure within the improved approach. The corresponding results for the Gegenbauer coefficients are presented. With this values we plot the

QCD-based $\eta$-meson energy spectrum in the inclusive $\Upsilon (1S) \to \eta g g g \to \eta + X$ decay in the leading-order perturbative QCD in the static-quark limit for the orthoquarkonium. Both the $\eta$- and $\eta^\prime$-meson energy spectra in the decays considered can be checked by the Belle Collaboration based on the existing and forthcoming data on $\Upsilon (1S)$-meson.