Remark

One can use formula (33) (resp. (34)) to calculate the $\beta_n$ (resp. the $u_n(x)$) by using the following lemma on entire series:

(1) Later we shall study series whose general terms are of the form $c_n z^n$ ($c_n \in \mathbf{C}, z \in \mathbf{C}$), which one calls entire series; in particular we shall see that when such a series is absolutely convergent for $z = z_0$ it is normally convergent for $|z| \leq |z_0|$. (2) It follows from the theory of entire series that when one of these series is absolutely convergent in a neighbourhood $V$ of 0, the other is absolutely convergent in a neighbourhood $W \subset V$ of 0.

emma. If two entire series $\sum_{n=0}^{\infty} c_n \lambda^n,\ \sum_{n=0}^{\infty} d_n \lambda^n$ are absolutely convergent for all $\lambda$ in a neighbourhood of $0$, and if $\sum_{n=0}^{\infty} c_n \lambda^n = \sum_{n=0}^{\infty} d_n \lambda^n$ for these values of $\lambda$, then $c_n = d_n$ for all integers $n \geqslant 0$.

If, by any procedure, one can find an entire series that converges to $\lambda^p / u(\lambda)$ on a neighbourhood of $0$, then the coefficients of this series are necessarily equal to the $\beta_n / n!$. Is this procedure that we shall apply in the examples that follow.