Hypergeometric 1F1 identity

Question

I have the following hypergeometric 1F1 function

\begin{equation} x{_1}F_1(a-3/2;2a-1;c x^2) \end{equation}

Is ther any way such that i can express this into something like $ {_1}F_1(a-1;2a-1;c x^2)$?

Answer 1

You can't have the direct form you want since the constant term for all hypergeometric functions is is non-zero.
The standard derivative works: The derivative calculation is also elementary. $\frac{d}{dx}\left(_{1}F_{1}\left(d;b;c\cdot x^{2}\right)\right)=\frac{d\cdot2\cdot c\cdot x}{b}{}_{1}F_{1}\left(d+1;b+1;c\cdot x^{2}\right) $
$\therefore$
$x\cdot_{1}F_{1}\left(a-3/2;2\cdot a-1;c\cdot x^{2}\right)=\frac{2\cdot a-2}{\left(a-5/2\right)\cdot2\cdot c}\cdot\frac{d}{dx}\left(_{1}F_{1}\left(a-5/2;2\cdot a-2;c\cdot x^{2}\right)\right)$