I am looking to re-express ${}_1F_1(n+\frac12,b,x)$ for $n$ a positive integer. Actually, so is b. I wrote:
\begin{equation} \begin{split} {}_1F_1(n + \frac12,b,x) & = 1 + \sum_{k=1}^\infty \frac{(2n+1)(2n+3)\ldots(2n+k-1)}{2^k b(b+1)\ldots(b+k-1)} \frac{z^k}{k!} \\ & = 1 + \sum_{k=1}^\infty\frac{2n(2n+1)(2n+2)\ldots(2n+2k-1)}{ n(n+1)(n+2)\ldots (n+k-1)b(b+1)\ldots(b+k-1)} \frac{\left(\frac z2\right)^k}{k!}\\ & =\ldots \end{split} \end{equation}
So, the bottom part can be reexpressed as $bn$ of the ${}_1F_1$ function. What can I do to the top part? I was thinking of $_{2}F_1$ but I am not sure what to do with the numerator (and not even sure it can be so expressed). So, I have been stuck for a while!
Is there some expression (and simpler form) for this? Many thanks for suggestions!