Question Number 67520 by mathmax by abdo last updated on 28/Aug/19
$${let}\:{f}\left({x},{z}\right)\:=\frac{{z}\:{e}^{{xz}} }{{e}^{{z}} −\mathrm{1}}\:\:\:\:\:\:\left({x}\:{and}\:{z}\:{from}\:{C}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}\left({x},{z}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{B}_{{n}} \left({x}\right)\frac{{z}^{{n}} }{{n}!} \\ $$$${with}\:{B}_{{n}} \left({x}\right)\:{is}\:{a}\:{unitaire}\:{polynome}\:{with}\:{degre}\:{n} \\ $$$${determine}\:{B}_{{n}} \left({x}\right)\:{interms}\:{of}\:{B}_{{n}} \left({number}\:{of}\:{bernoulli}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{B}\:_{{n}} ^{'} \left({x}\right)={nB}_{{n}−\mathrm{1}} \left({x}\right) \\ $$$${B}_{{n}} \left({x}+\mathrm{1}\right)−{B}_{{n}} \left({x}\right)\:={nx}^{{n}−\mathrm{1}} \\ $$$${prove}\:{that}\:{f}\left({x},{z}\right)={f}\left(\mathrm{1}−{x},−{z}\right)\:\:{and}\:{B}_{{n}} \left(\mathrm{1}−{x}\right)\:=\left(−\mathrm{1}\right)^{{n}} \:{B}_{{n}} \left({x}\right) \\ $$