If a worshipper catches the congregation in the 'Asr prayer while they are bowing, should he recite the opening supplication after the تكبيرة الإحرام (Takbirat al-Ihram) and then bow, or should he bow immediately without reciting the supplication? And is it permissible to bow directly without the تكبيرة الإحرام?

Scholars have unanimously agreed that the opening supplication (du'a al-istiftah) is not among the pillars or obligations of prayer. Therefore, if a worshipper fears missing the bowing (ruku'), he is not required to recite it. It is obligatory to begin prayer with the opening takbir (takbirat al-ihram (definition)) while standing, then to say the takbir while descending for bowing without reciting the opening supplication or any other recitation. However, if someone bows with the imam without performing the opening takbir, their prayer is invalid.