While reading a research article, I come accros this expression: $$ {\sum\limits_{m = 0}^\infty {\left( {\frac{{p - \sqrt {{p^2} - {\alpha ^2}} }}{{\alpha \beta }}} \right)} ^m} = \frac{\lambda }{{{\beta ^{m + 1}}}}\left[ {{I_{m - 1}}\left( {\alpha \left( t \right)} \right) - {I_{m + 1}}\left( {\alpha \left( t \right)} \right)} \right]{e^{ - \left( {\lambda + \mu } \right)t}},$$ where
- $p = s + \lambda + \mu$ ,
- $\alpha = 2\sqrt {\lambda \mu } $ ,and
- $\beta = \sqrt {\frac{\lambda }{\mu }} $.
Please help me how they arrived this expression