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  3. 1) define a sequence of polynomials h n (x ) by h

1) Define a sequence of polynomials H n (x ) by H 0 (x )=1,...

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Advance Math

1) Define a sequence of polynomials H n (x ) by H 0 (x )=1, H 1(x )=2 x , and for n>1 by H n+1 (x )=2 x H n (x )−2 n H n−1 (x ). These polynomials are called Hermite polynomials of degree n.Calculate the first 7 Hermite polynomials of degree less than 7.You can check your results by comparing them to the list of Hermitepolynomials on wikipedia (physicist's Hermite polynomials).

2) Use the power series method to solve the differentialequation y ' '−2 x y '+λ y=0 where λ is an arbitrary constant.Verify that you get two independent solution y1, y2 by choosinga0=1, a1=0 and a0=0 , a1=1 . Show that the series expansion for oneof the two solutions will terminate resulting in a polynomialsolution when λ is chosen to be a positive even integer, λ=2,4,6,8,10 ,12 ,14 ,.... Rescale the polynomial solution so itstarts with 2 n x n + lower powers of x , n=λ/2. Calculate the listof polynomials obtained this way and compare them to your solutionof problem 1)

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