Generating function

Question

I need an reference or an idea for find the following generating function $$\sum_{n=0}^{+\infty}\frac{t^n}{n!}\ _1F_1(2n+b; c; x)$$ where $_1F_1$ the Kummer's function and $|t|<1$. Thanks

Answer 1

Hint:

$\sum\limits_{n=0}^\infty\dfrac{t^n}{n!}~_1F_1(2n+b;c;x)$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{(2n+b)^{(k)}x^kt^n}{c^{(k)}k!n!}$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{\Gamma(2n+k+b)x^kt^n}{\Gamma(2n+b)c^{(k)}k!n!}$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{\Gamma(2n+2k+b)x^{2k}t^n}{\Gamma(2n+b)c^{(2k)}(2k)!n!}+\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{\Gamma(2n+2k+b+1)x^{2k+1}t^n}{\Gamma(2n+b)c^{(2k+1)}(2k+1)!n!}$