Let's consider the series: $$ F(t) = \sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{\infty} \frac{(-b)^k(-a)^n\binom{n+k}{k} t^{2n+k(2-\alpha)}}{\Gamma(2n+k(2-\alpha)+2)} $$
where $a,b$ are positive reals, $0 < \alpha < 1$, and $\Gamma(z)$ is the gamma function.
I would like to numerically compute the sum $F$. I replace the doubly infinite sum by a double finite sum in the following way:
$$ \sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{\infty}\left(\ldots\right) \approx \sum\limits_{i=0}^{N}\sum\limits_{{n+k = i}\atop{0\leq n,k\leq i}} \left(\ldots\right) \ \ \ (\star)$$
where bound $N$ is set in such a way, that $\left|\frac{S_{N+1}-S_{N}}{S_{N+1}}\right|<\varepsilon$ ($S_j$ - sum from right hand path of the ($\star$) for $N=j$; $\varepsilon$ - some small number, for example $\varepsilon = 10^{-6}$).
Unfortunately there is something wrong with my approach. For example for $a=1$, $b=1$, $\alpha=0.5$ and $0\leq t \leq 30$ I get following graph:
Definitely, there is something wrong for $t\geq 25$.
My question: How to compute approximation of $F(t)$ correctly also for $t\geq 25$?
Code (in this way I define function $F(t)$):
F[t_, α_] := Block[{convergence, N1, S, SN, tab, re, i},
N1 = 0;
S = 1;
convergence = False;
While[! convergence && N1 <= 1000,
N1++;
(* ponizej k = i, n = N1-i *)
tab = Table[((-v0)^i (-w0^2)^(N1 - i) Binomial[N1, i] t^(
2 (N1 - i) + i (2 - α)))/
Gamma[2 (N1 - i) + i (2 - α) + 2], {i, 0, N1}];
SN = S + Sum[tab[[i]], {i, 1, Length[tab]}];
re = Abs[(S - SN)/S];
If[re < 10^-6, convergence = True];
S = SN;
];
Return[{S, N1, re}];
]
