Did Jainism influence Buddhism or did Buddhism influence Jainism?

Or if they influence each other, which came first?

Buddhism influenced Jainism

Both are telling the same truth in different ways.
Truth cannot influence itself.

Both were in a way are an off-shoot of Sanatana Dharma (Vedic Hindu religion). Both are some what different and Janism is stricter than Buddhism. Buddhists and Jains have many traditions influenced by Hindus, the difference being Buddhists\Jains dint worship "Nirguna Brahman". Buddha was a Hindu. He did not renounce Hindu religion. Their scriptures are close to Hindu scriptures like Vedas and Upanishads.

There are no such religions as Buddhism and Jainism. Many people wrongly assume that these are independent religions in their own right but they are only departments within Hinduism. It is more correct to view Buddhism and Jainism as affiliates of Hinduism or an inseparable part of Hinduism rather than individual religions. The same logic applies to Sikhism.

Both the religions are branches of periodic and prepaying Shraman tradition which existed in Indus valley civilization also. Bapp, the paternal uncle of Goutam Buddha was a Jain monk. According to Buddhist texts like 'Majjhimnikay', Goutam Buddha himself practiced Jain monk hood for 6 months. Then he established his own way.

The parents of both Buddha and Mahavira were followers of the tradition of Parshwanath, the 23rd Teerthankar of Jain order.

So it is clear that the main principles of both religion are the same because of their being branches of same prior Shraman tradion. They have not influenced each other much, but they influnced Hinduism. Today's Hinduism is nothing but a mixed traditon of Jainism, Buddhism and tradion of other aboriginals. It has nothing to do with Vedic traditions. Vedic religion is a separate religion from Hinduism and the non-Bramhin Hindus know nothing about Vedas.

Arya Nagarjuna, who revived the Prajnaparamita, refuted the Jains philosophical position that a thing is produced from both itself and other. He did this by refuting the Samkhya's and the Autonomists who posited that a thing is produced from self and other, respectively. That a thing could be produced from both after each, self and other, had been refuted made absolutely no sense! Further, the Nihilists were refuted who were convinced that there was no previous life and no future life and said that a thing had no cause. These were all Indian Hindu philosophical schools that were defeated by the great master Arya Nagarjuna, the founder of the Madhyamika-Prasangika, the highest philosophical tenet system that identified the lack of true establishment of not only the self, but also the lack of true establishment of phenomenon saying that not only is the object perceived by a mental consciousness empty of inherent existence, but the consciousness that apprehends the object and imputes a name and label on that object is empty as well. In the context of conventional and ultimate truth - things exist, but not inherently - they are a collection that is named and can perform a function. Table, for instance. So, in my opinion, philosophically, they are mutually exclusive. One can not ride two horses at once, and sooner or later, you have to get down to brass tacks, the metaphysics underlying any philosophy. How do you define a thing?