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Question Number 33592 by abdo imad last updated on 19/Apr/18
$${let}\:{f}\left({x}\right)\:={e}^{−{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)\:=\:{p}_{{n}} \left({x}\right).{e}^{−{x}^{\mathrm{2}} } \:\:\:{where}\:{p}_{{n}} {is}\:{a}\:{polynome} \\ $$$${with}\:{deg}={n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\geqslant\mathrm{1}\: \\ $$$${p}_{{n}+\mathrm{1}} \left({x}\right)\:+\alpha\left({x}\right){p}_{{n}} \left({x}\right)\:+\beta\left({n}\right){p}_{{n}−\mathrm{1}} \left({x}\right)\:=\mathrm{0}\:\:{find}\:\alpha\:{and}\:\beta \\ $$$$\left.\mathrm{3}\right){calculate}\:{p}_{\mathrm{0}} ,{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} ,{p}_{\mathrm{3}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{p}_{{n}} ^{''} \:\left({x}\right)\:{interms}\:{of}\:{p}^{'} \left({x}\right)\:{and}\:{p}_{{n}} \left({x}\right). \\ $$