MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Are there any p-adic techniques that can be applied to the 2F1 hypergeometric function?

For e.g. I'm interested in which values this function converges p-adically.

share|cite|improve this question

The Gauss hypergeometric function is the main example in the theory of p-adic differential equations. See

K. S. Kedlaya, p-Adic Differential Equations, Cambridge University Press, 2010,

for the general theory. There were also two books by Dwork, almost completely devoted to ${}_2F_1$ (its p-adic theory is much more complicated than the classical one):

B. Dwork, Generalized hypergeometric functions. Oxford: Clarendon Press, 1990.

B. Dwork, Lectures on p-adic differential equations, Springer, 1982.

It is easy to check local p-adic convergence for the hypergeometric series, but to study and even correctly define its analytic continuation properties one needs subtle analytic and algebraic techniques.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.