I have function $f(t)$ defined as follow: $$ f(t) = \sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{\infty} \frac{(-b)^n(-a)^k\binom{n+k}{k}} {\Gamma(k(\alpha-\beta)+(n+1)\alpha-\gamma)} t^{k(\alpha-\beta)+n\alpha} $$ where $a,b,\alpha,\beta,\gamma$ -- some real constants such that $\alpha\geq\beta$, $\alpha>\gamma$.
I'd like to compute values of this function numerically to draw its graph. In my case constants $a$ and $b$ are big ($a,b\approx 10^8$), $\alpha,\beta$ and $\gamma$ are small ($\alpha = 2$, $0<\beta<1$, $\gamma = -1$).
At first approach I tried to cut infinite sum at some level, however sequence $a_{kn}$ for given $t$ $$ a_{kn} =\frac{(-b)^n(-a)^k\binom{n+k}{k}} {\Gamma(k(\alpha-\beta)+(n+1)\alpha-\gamma)} t^{k(\alpha-\beta)+n\alpha} $$ converges to zero very slow and I have to take too many terms in sums to obtain result in reasonable time.
My question: How to compute such sums numerically with good accuracy and in reasonable time? (I have experience in using softwere for symbolic computation, but for example in Mathematica for $a$ and $b$ in given level command NSum does not work for such big $a,b$)