I want to compute the following integrals:
$$ \int y^{a} e^{\frac{1}{2}y}M_{k,m}(y)dy $$ where a is an arbitrary constant and $M_{k,m}$ is a whittaker function of the first kind.
I already know that it is possible to compute cases when $a=m-\frac{1}{2},-m-\frac{1}{2}, m+\frac{1}{2}, -m+\frac{1}{2}, k,$or $-k$.
by using the derivatives of Whittaker functions.
My question is how to formulate when $a$ is a arbitrary constant. I know some relevant reference but i couldn't get that reference in my library and google.
Relevant references are :
A.Prudnikov et al. "Integrals Series: More special functions, vol. III of integrals and Series" pp 105--109.
Thank you for your answer.
Wolfram alpha gives the result $$\frac{2 y^{a+m+\frac{3}{2}} \, _2F_2\left(a+m+\frac{3}{2},-k+m+\frac{1}{2};a+m+\frac{5}{2},2 m+1;y\right)}{2 a+2 m+3}.$$