I think a fun project would be for someone to define hypergeometric functions and prove some basic properties.
Eventually we would want the generalised $${}_pF_q$$ but I think even just starting with Gauss's $${}_2F_1$$ would be a great addition and probably important enough to deserve its own definition. I can imagine developing the theory along the following lines:
- define $${}_2F_1$$
- prove it is analytic in the disc
- prove it satisfies the hypergeometric differential equation
Possibly most of this effort would be developing API for power series (I'm not sure just how much we have, though we do at least have some versions of the ratio test).
More ambitious goals might be:
- prove that it can be analytically continued (taking the usual branch cut $$[1, \infty)$$)
- derive Schwarz's list (i.e., classify parameters for which $${}_2F_1$$ has finite monodromy)
I think a fun project would be for someone to define hypergeometric functions and prove some basic properties.
Eventually we would want the generalised$${}_pF_q$$ but I think even just starting with Gauss's $${}_2F_1$$ would be a great addition and probably important enough to deserve its own definition. I can imagine developing the theory along the following lines:
Possibly most of this effort would be developing API for power series (I'm not sure just how much we have, though we do at least have some versions of the ratio test).
More ambitious goals might be: