Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables following an \(\operatorname{Gamma}(\alpha, \beta)\) density.

a. Find the value of \(\tilde{\lambda}\) that is the solution to \(n c^{\prime}(\tilde{\lambda})=x\).

b. Find \(c(\tilde{\lambda})\) and \(c^{\prime \prime}(\tilde{\lambda})\).

c. Derive the saddlepoint expansion for \(f_{n}(x)\).