Were the Buddhist texts written by Sākyamuni himself?
佛经是释迦牟尼亲自写的吗？

No, they were recited and recorded by his disciples after the Buddha's death. In the year of the Buddha's Parinibbāna, his five hundred disciples headed by Ven. Mahākassapa held an assembly at Saptaparṇa Cave (P. Sattapaṇṇī-guhā) near Rājagaha to compile and edit the Buddha's teachings for posterity. At the assembly Ven. Ānanda recited the Suttas preached by the Buddha, Ven. Upāli recited the Vinaya established by the Buddha, and Ven. Mahākassapa recited, and later supplemented, the Abhidamma which is an exposition and study of the Buddhist creeds. The Sutta, Vinaya and Abhidamma comprise Tipiṭaka. The word Piṭaka originally meant basket for containing things. The compilation of Sutta, Vinaya and Abhidamma into Tipiṭaka is something like the designation of Jing (classics), Shi (history), Zi (academic schools) and Ji (miscellany) as the "Four Treasures" in China. This Buddhist council was termed the First Saṃgīti. Saṃgīti is generally rendered as "Jieji" in Chinese, while its original meaning in Sanskrit or Pāli is "saṅgha meeting". The ancient Chinese translators used the word "Jieji" to mean "recital", which implies both the collection of Dhamma, and the assembly of people. The Tipiṭaka was not written down at that time, so it was passed on by oral recital.
不是，是佛逝世后，他的弟子记诵出来的。佛逝世的那一年，佛的弟子，以摩诃迦叶为首的五百人集会在王舍城外的七叶窟，将佛一生所说的言教结集起来，以传后世。当时由阿难陀诵出佛所说的经（Sutta）；由优波离诵出佛所制的僧团戒律（Vinaya）；由摩诃迦叶当时诵出，后来又补充结集的关于教理的解释和研究的论著（Abhidhamma）。经、律、论为三藏Tipiṭaka。藏的原语Piṭaka，是一种可以盛放东西的竹箧。把经、律、论分为三藏，同我国把经、史、子、集分为四库有差不多的意思。这一次结集称为第一结集。照我国通常解释，结集两个字含有编辑的意义。但这个字的梵文Saṃgīti（巴利语同）却是僧众大会的意思。古代译师用结集二字是含有“会诵”的意思，一方面固然着重在法的结集，同时也包含着人的结集的意思在内。这时还没有用文字记录，只凭口头传诵。