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.
It would be helpful to learn about the specific strengths of your language, since good examples should illustrate its strengths (yes, this is more a meta-answer, but maybe exactly what you should be looking for).
The standard example is printing "Hello World"; everything else depends on the language and its features. E.g. you might want to demonstrate simple database queries, easy list processing, recursion, string manipulation, logic deduction, complex mathematics, ...