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Question Number 33592 by abdo imad last updated on 19/Apr/18

l e t f (x) = e^{- x^{2}}

1) p r o v e t h a t f^{(n)} (x) = p_{n} (x) . e^{- x^{2}} w h e r e p_{n} i s a p o l y n o m e

w i t h d e g = n

2) p r o v e t h a t \forall n ⩾ 1

p_{n + 1} (x) + α (x) p_{n} (x) + β (n) p_{n - 1} (x) = 0 f i n d α a n d β

3) c a l c u l a t e p_{0}, p_{1}, p_{2}, p_{3}

4) c a l c u l a t e p_{n}^{^{″}} (x) i n t e r m s o f p^{^{'}} (x) a n d p_{n} (x) .