Rule 1: 1. اَللّٰهُمَّ: 'O God'
اَللّٰهُمَّ is a special vocative of اَللّٰهُ: the final ـمَّ replaces the vocative particle, so it is never used with يَا. It stands at the head of a prayer and is followed by a request in the imperative, by لَا + jussive, or by a further epithet in the vocative (accusative) such as رَبَّنَا. It may also open a sentence of praise: اَللّٰهُمَّ لَكَ الْحَمْدُ.
اَللّٰهُمَّ اغْفِرْ لَهُ وَارْحَمْهُ.
allāhumma ghfir lahu wa-rḥamhu.
O God, forgive him and have mercy on him.
اَللّٰهُمَّ افْتَحْ عَلَيْهِ.
allāhumma ftaḥ ʿalayhi.
O God, open (understanding) to him.
Said to a pupil who stumbles in recitation.
اَللّٰهُمَّ ارْزُقْنِي وَإِيَّاهُ.
allāhumma rzuqnī wa-iyyāhu.
O God, provide for me and for him.
إِيَّاهُ: the independent object pronoun after وَ.
اَللّٰهُمَّ لَكَ الْحَمْدُ.
allāhumma laka l-ḥamdu.
O God, to You belongs praise.