Question c) The Hermite polynomials Hn(x) are defined by the generating function: th = ΣΗ() n! 2 g(x, t) = exp(-+? + 2tx) = Ï H.(x) = n=0 Expand exp(-ť? + 2tx) as a Maclaurin series and show that the first two Hermite polynomials are: H.(x) = 1 and H1(x) 2x = Hint: yn y2 you ey IM8 = 1+ y + + 2! n=on! 3! (8 + (4+5) + (4+4) marks)
XOIHJT The Asker · Advanced Mathematics
Transcribed Image Text: c) The Hermite polynomials Hn(x) are defined by the generating function: th = ΣΗ() n! 2 g(x, t) = exp(-+? + 2tx) = Ï H.(x) = n=0 Expand exp(-ť? + 2tx) as a Maclaurin series and show that the first two Hermite polynomials are: H.(x) = 1 and H1(x) 2x = Hint: yn y2 you ey IM8 = 1+ y + + 2! n=on! 3! (8 + (4+5) + (4+4) marks)
More Transcribed Image Text: c) The Hermite polynomials Hn(x) are defined by the generating function: th = ΣΗ() n! 2 g(x, t) = exp(-+? + 2tx) = Ï H.(x) = n=0 Expand exp(-ť? + 2tx) as a Maclaurin series and show that the first two Hermite polynomials are: H.(x) = 1 and H1(x) 2x = Hint: yn y2 you ey IM8 = 1+ y + + 2! n=on! 3! (8 + (4+5) + (4+4) marks)
Community Answer
GAAY8W
c){:[g(x","t)=exp(-t^(2)+2tx)=sum_(n=0)^(oo)H_(n)(x)(t^(n))/(n!)],[:.quade^(y)=1+y+(y^(2))/(2!)+(y^(3))/(3!)+dots]:}by eq. (1){:[=>e^((-t^(2)+2tx)){:=e^(t(2x-t))],[=1+t(2x-t)+(t^(2)(2x-t)^(2))/(2b)+(t^(3 ... See the full answer