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14 Uvod1.1.5 Hermiteovi polinomi i funkcijeDefinicija 1.1.32 Hermiteov polinom (ermitski polinom) n-tog reda, n ∈N 0 je definisan sah n (x) = (−1) n e x2 d n x22 (e− 2dxn ). (1.32)Prvih nekoliko ermitskih polinoma su: h 0 (x) = 1, h 1 (x) = x,h 2 (x) = x 2 − 1, h 3 (x) = x 3 − 3x, h 4 (x) = x 4 − 6x 2 + 3 . . . itd.Teorema 1.1.20 Hermiteovi polionomi zadovoljavaju rekurentne veze:1. h n+1 (x) − xh n (x) + nh n−1 (x) = 0,2. h ′ n(x) = nh n−1 (x),3. h ′′ n(x) − xh ′ n(x) + nh n (x) = 0.Teorema 1.1.21 Familija { 1 √n!h n (x) : n ∈ N 0 } čini ortonormiranu bazuprostora L 2 (R, dµ), gde je dµ = 1 √2πe − x22 Gaussova mera.Teorema 1.1.22 Generativna funkcija Hermiteovih polinoma jee tx− 1 2 t2 =∞∑n=0t nn! h n(x). (1.33)Teorema 1.1.23 Svaki polinom nad poljem realnih brojeva se može prikazatipreko ermitskih polinoma i obratno, preko identiteta:x n =[n/2]∑k=0( n2k)(2k − 1)!!h n−2k (x), (1.34)[n/2]∑( ) nh n (x) = (−1) k (2k − 1)!!x n−2k . (1.35)2kk=0Teorema 1.1.24 Hermiteovi polinomi zadovoljavaju identitete:1. h n (x + y) = ∑ nk=0( nk)hn−k (x)y k ,2. h n (ax) = ∑ [n/2]k=0 (−1)k( n2k)(2k − 1)!!a n−2k (1 − a 2 ) k h n−2k (x),3.1 √2π∫ ∞−∞ h n(x + y)e − x22 dx = y n ,