On the Padé table for \(e^{x}\) and the simple continued fractions for \(e\) and \(e^{L/M}\) (Q1039644)

The Padé table of the formal power series \[ f(x)=\sum_{k=0}^\infty c_kx^k,\quad c_0\neq0, \tag{1} \] is an infinite two dimensional array of irreducible rational functions \[ P_{n,m}(x)=\frac{A_{n,m}(x)}{B_{n,m}(x)}=\frac{\alpha_0+\alpha_1x+ \alpha_2x^2+\cdots+\alpha_mx^m}{\beta_0+\beta_1x+\beta_2x^2+\cdots+\beta_nx^n},\quad m, n>0,\tag{2} \] in each of which the coefficients are such that the expansion of \(P_{n,m}(x)\) in powers of \(x\) matches that of \(f(x)\) as far as possible. The power series and its associated Padé table are said to be normal if \(P_{n,m}(x)=\sum_{k=0}^{m+n} c_kx^k+ \text{higher order terms}\), in which case every element of the table exists and is different from any other element. There are several methods for transforming a normal series into its Padé table, including variations of the quotient difference algorithm, techniques which exploit the close connection between the Padé table and various continued fraction expansions that correspond to the series (1). In this paper, one such algorithm is developed for transforming two series expansions into two point Padé approximations. A recently observed connection between some Padé approximants for the exponential series and the convergents of the simple con\-ti\-nu\-ed fraction for \(e\) is established, leading to an alternative proof of the latter. Similar results for the simple con\-ti\-nu\-ed fraction \(e^2\), \(e^{1/M}\) and \(e^{2/M}\), when \(M\) is a natural number greater than one, are derived.