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

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