So I had a look at
E. Borel: Leçons sur les séries divergentes (1901)
In chapter V, page 156 he explains the Mittag-Leffler expansion.
Mostly I fighted my way through the text with an online translator.
I found the following interesting formulas, where \( \phi \) is the function we search the expansion of:
\( g_n(x)=\sum_{\lambda_1=0}^{n^{2n}}\sum_{\lambda_2=0}^{n^{2n-2}}
\dots\sum_{\lambda_n=0}^{n^2} \frac{\phi^{(\lambda_1+\dots+\lambda_n)}(0)}{\lambda_1!\dots\lambda_n!} \left(\frac{x}{n}\right)^{\lambda_1+\dots+\lambda_n}
\)
\( G_0(x)=g_0(x)=\phi(0) \)
\( G_n(x)=g_n(x)-g_{n-1}(x) \) for \( n>0 \),
\( \phi(x)=\sum_{n=0}^\infty G_n(x) \)
E. Borel: Leçons sur les séries divergentes (1901)
In chapter V, page 156 he explains the Mittag-Leffler expansion.
Mostly I fighted my way through the text with an online translator.
I found the following interesting formulas, where \( \phi \) is the function we search the expansion of:
\( g_n(x)=\sum_{\lambda_1=0}^{n^{2n}}\sum_{\lambda_2=0}^{n^{2n-2}}
\dots\sum_{\lambda_n=0}^{n^2} \frac{\phi^{(\lambda_1+\dots+\lambda_n)}(0)}{\lambda_1!\dots\lambda_n!} \left(\frac{x}{n}\right)^{\lambda_1+\dots+\lambda_n}
\)
\( G_0(x)=g_0(x)=\phi(0) \)
\( G_n(x)=g_n(x)-g_{n-1}(x) \) for \( n>0 \),
\( \phi(x)=\sum_{n=0}^\infty G_n(x) \)
