Let's consider following double power series: $$\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{\infty} a_{nk}t^{f(n,k)}$$ where $f(n,k)$ - some linear funkction of $n$ and $k$ (increasing in $n$ and $k$ directions), $t$ - real number.
1) How to define radius of convergence of such series?
2) How to compute radius of convergence?
3) How to compute radius of convergence for $a_{nk} = \frac{(-a)^k(-b)^n{ n+k \choose k}}{\Gamma(2n+(2-\alpha)k+2)}$ and $f(n,k) = 2n+(2-\alpha)k$, where $a,b>0$, $0<\alpha<1$, $\Gamma$ - Gamma function?