Expansion of hypergeometric function for large arguments

Question

Does anybody know of an asymptotic expansion of $${}_2 F_1 \left( {a, \ b+\lambda\atop c+\lambda}; z \right)$$ for large $\lambda$ and $z \to 1^-$? Alternative just for bounded $z$. I found a big literature on this topic but the only expansion I found in this particular case is for large $z$. I expect some expansion with Kummer U functions, but any expansion is welcomed.

Answer 1

For $z$ bounded away from 1 and < 1 you can asymptotically expand the ratio of Pochhammer symbols: $$ \sum_{k=0}^\infty \frac{(a)_k}{k!} \frac{(b+\lambda)_k}{(c+\lambda)_k} z^k \sim \sum_{k=0}^\infty \frac{(a)_k}{k!} z^k \Big(1 + (b-c)\frac{k}{\lambda}+\,...\Big)$$ $$ = (1-z)^{-a} \Big( 1+ \frac{(b-c)}{1-z} \, \frac{a\,z}{\lambda} + ... \Big) $$

For $z$ very close to one you can probably use a hypergeometric identity that takes $z \to 1-z.$ There will be a region such that $z$ approaches 1 as a function of $\lambda$ in which simple functions won't work, but I'd try the simple things first.