Is there a standard system example for writing papers which introduce a newly developed programming language? I am writing a paper that describes a programming language that I am developing as part of my PhD. I think that writing the paper using an example can greatly improve how the paper is understood. Is there a "standard" example that is used in programming languages paper? A set of examples?
The biggest problem that you face when writing a programming language paper is explaining why the world needs another programming language, rather than a library or an API or something of the sort.

I would thus recommend skipping "standard" examples such as "Hello world" and recursive factorials entirely, and instead going straight to a set of minimal examples that illustrate the value of the new language.  Only in the case of a particularly extreme langauge will the standard examples be necessary to help understand the scientific concepts.

Note that documenting a programming language is an entirely different matter.