Éléments de 1ère espèce
\[\begin{align*} \FFF 1/(x^2-1)^5 =& \FFF1/32(x-1)^{5} -\FFF5/64(x-1)^{4} +\FFF15/128(x-1)^{3} -\FFF35/256(x-1)^{2} +\FFF35/256({x-1}) \\ -&\FFF35/256({x+1}) -\FFF35/256(x+1)^{2} -\FFF15/128(x+1)^{3} -\FFF15/64(x+1)^{4} -\FFF1/32(x+1)^{5} \cr \FFF (x^2+1)^{2}/(x-1)^{6} =& \FFF4/(x-1)^{6} + \FFF8/(x-1)^{5} + \FFF8/(x-1)^{4} + \FFF4/(x-1)^{3} + \FFF1/(x-1)^{2} \cr \FFF x^3+x+1/x^4(x-1)^{3} =& -\FFF1/x^4 -\FFF4/x^3 -\FFF9/x^2 -\FFF17/x +\FFF3/(x-1)^{3} -\FFF8/(x-1)^{2} +\FFF17/{{x-1}} \cr \FFF (x^2-x+1)^2/x^2(x-1)^{2} =& 1 + \FFF1/x^2 + \FFF1/(x-1)^{2} \cr \FFF x^2/(x^2-1)^2 =& \FFF1/4(x-1)^{2} + \FFF1/4({x-1}) + \FFF1/4(x+1)^{2} + \FFF1/4({x+1}) \cr \end{align*}\]
Du type \(x^2+1\)
\[\begin{align*} \FFF x^2/(x^2+1)^2 =& \FFF -1/(x^2+1)^{2} + \FFF 1/{{x^2+1}} \cr \FFF x/(x^4-1)^2 =& \FFF1/16(x-1)^{2} -\FFF1/8({x-1}) - \FFF1/16(x+1)^{2} -\FFF1/8({x+1}) + \FFF x/4(x^2+1)^{2} + \FFF x/4({x^2+1}) \cr \FFF x/({x-1})(x^2+1)^{2} =& \FFF 1/4({x-1}) + \FFF 1-x/2(x^2+1)^{2} - \FFF x+1/4({x^2+1}) \cr \FFF x^6/(x^2+1)^{2}(x+1)^{2} =& 1 +\FFF1/4(x+1)^{2} - \FFF1/{{x+1}} + \FFF x/2(x^2+1)^{2} - \FFF {x+1/4}/{{x^2+1}} \cr \FFF x^6/({x^2+1})(x-1)^{3} =& x + 3 + \FFF x-1/4({x^2+1}) + \FFF 1/2(x-1)^{3} + \FFF 5/2(x-1)^{2} + \FFF 19/4({x-1}) \cr \end{align*}\]
Du type \(x^2+x+1\)
\[\begin{align*} \FFF x/x^4+x^2+1 =& \FFF 1/2(x^2-x+1) - \FFF 1/2(x^2+x+1) \cr \FFF x^4+1/x^4+x^2+1 =& 1 + \FFF x/2(x^2+x+1) - \FFF x/2(x^2-x+1) \cr \FFF x^4+1/x^2(x^2+x+1)^{2} = & \FFF1/x^2 -\FFF2/x -\FFF1/(x^2+x+1)^{2} + \FFF2x+2/{{x^2+x+1}} \cr \FFF 3x^5-5x^4+4x^2-11x+1/(x^2+x+1)^{6} =& -\FFF 23x+6/(x^2+x+1)^{6} + \FFF 13x+18/(x^2+x+1)^{5} + \FFF 3x-11/(x^2+x+1)^{4} \cr \end{align*}\]
Autres éléments de 2ème espèce
\[\begin{align*} \FFF x^8/x^6-1 =& x^2 + \FFF1/6 \biggl( \FFF1/x-1 -\FFF1/x+1 +\FFF2x+1/x^2+x+1 -\FFF2x-1/x^2-x+1 \biggr) \cr \FFF 1/x^4+1 =& \FFF1/2\sqrt2 \biggl( \FFF x+\sqrt2/x^2+x\sqrt2+1 - \FFF x-\sqrt2/x^2-x\sqrt2+1 \biggr) \cr \FFF x/x^4+1 =& \FFF 1/2\sqrt2 \biggl( \FFF 1/x^2-x\sqrt2+1 - \FFF 1/x^2+x\sqrt2+1 \biggr) \cr \FFF 1/x^5+1 =& \FFF 1/5({x+1}) - \FFF1/5 \biggl( \FFF {\omega x-2}/{x^2-\omega x+1} + \FFF {\omega'x-2}/{x^2-\omega'x+1} \biggr),\quad \omega = \FFF1+\sqrt5/2 ,\omega' = \FFF1-\sqrt5/2 \cr \end{align*}\]
Racines de l’unité
\[\begin{align*} \FFF x^n+1/x^n-1 =& 1 + 2\sum_{k=0}^{n-1} \FFF \omega^k/n(x-\omega^k) ,\quad \omega = e^{2i\pi/n}\cr \FFF 1/x^n-1 =& \sum_{k=1 ; 2k \ne n}^{n-1} \FFF 2x\cos\alpha_k-2/n(x^2-2x\cos\alpha_k+1) + \FFF1/n(x-1) \ \left[ - \FFF1/n(x+1) \text{ si n est pair} \right], \quad \alpha_k = \FFF2k\pi/n \cr \sum_{k=0}^{n-1} \FFF 1/x-\omega^k =& \FFF nx^{n-1}/x^n-1 ,\quad \omega = e^{2i\pi/n}\cr \sum_{k=0}^{n-1} \FFF 1/(x-\omega^k)^2 =& \FFF nx^{2n-2}+n(n-1)x^{n-2}/(x^n-1)^2 ,\quad \omega = e^{2i\pi/n}\quad(\text{d{é}riv{é}e})\cr \end{align*}\]
Polynômes de Tchebychev
\[\begin{align*} \FFF 1/{\cos(n\arccos x)} =& \FFF1/n \sum_{k=0}^{n-1} \FFF (-1)^k\sin\beta_k/x-\cos\beta_k ,\quad \beta_k = \FFF(2k+1)\pi/2n \cr \tan(n\arctan x) =& \FFF1/n \sum_{k=0 ; 2k\ne n-1}^{n-1} \FFF1/\cos^2\beta_k(\tan\beta_k-x) \left[ + \FFF x/n \ \text{si n est impair} \right],\quad \beta_k = \FFF(2k+1)\pi/2n \cr \end{align*}\]
Divers
\[\begin{align*} \FFF x^{2n}/{(x^2+1)^{n}} =& \sum_{k=0}^n \FFF (-1)^kC_n^k/{(x^2+1)^{k}} \cr \FFF 1/(x^2-1)^n =& \sum_{k=0}^{n-1} \FFF \Gamma_n^k/2^{n+k} \biggl( \FFF (-1)^k/(x-1)^{n-k} + \FFF (-1)^n/(x+1)^{n-k} \biggr) \cr \FFF 1/{(x^2+1)^{n}} = & \sum_{k=0}^{n-1} \FFF (-1)^n\Gamma_n^k/2^{n+k} \biggl( \FFF {i^{k+n}}/(x-i)^{n-k} + \FFF {(-i)^{k+n}}/(x+i)^{n-k} \biggr) \cr \FFF n!/(x+1)(x+2)\dots(x+n) =& \sum_{k=1}^n \FFF (-1)^{k-1}kC_n^k/x+k \cr \FFF x^2/x^4-2x^2\cos\alpha+1 =& \FFF1/4\cos(\alpha/2) \biggl(\FFF x/x^2-2x\cos(\alpha/2)+1 - \FFF x/x^2+2x\cos(\alpha/2)+1 \biggr), \quad \alpha\not\equiv0(\mathrm{mod}\,\pi)\cr \end{align*}\]