Padé approximant related to the Wallis formula
AbstractBased on the Padé approximation method, in this paper we determine the coefficients aj$a_{j}$ and bj$b_{j}$ such that 
 π=((2n)!!(2n−1)!!)2{nk+a1nk−1+⋯+aknk+1+b1nk+⋯+bk+1+O(1n2k+3)},n→∞,$$ \pi= \biggl(\frac{(2n)!!}{(2n-1)!!} \biggr)^{2} \biggl\{ \frac {n^{k}+a_{1}n^{k-1}+\cdots+a_{k}}{n^{k+1}+b_{1}n^{k}+\cdots+b_{k+1}}+O \biggl(\frac{1}{n^{2k+3}} \biggr) \biggr\} ,\quad n\to\infty, $$ where k≥0$k\geq0$ is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results.
New representations of Padé, Padé-type, and partial Padé approximants
Pade approximants are rational functions, with a denominator which does not vanish at zero, and whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different barycentric forms, and under a partial fraction form, depending on free parameters. According to the choice of these parameters, Pade-type approximants can be obtained under a barycentric or a partial fraction form.
Some Representations of pi
In this paper we consider a particular integral from which we may develop identities for pi.