Express 4x⁴+8x³+4x²-x-3 in terms of hermite polynomial
Answers
Answer:
The Hermite polynomials H_n(x) are set of orthogonal polynomials over the domain (-infty,infty) with weighting function e^(-x^2), illustrated above for n=1, 2, 3, and 4. Hermite polynomials are implemented in the Wolfram Language as HermiteH[n, x].
The Hermite polynomial H_n(z) can be defined by the contour integral
H_n(z)=(n!)/(2pii)∮e^(-t^2+2tz)t^(-n-1)dt,
(1)
where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).
The first few Hermite polynomials are
H_0(x) = 1
(2)
H_1(x) = 2x
(3)
H_2(x) = 4x^2-2
(4)
H_3(x) = 8x^3-12x
(5)
H_4(x) = 16x^4-48x^2+12
(6)
H_5(x) = 32x^5-160x^3+120x
(7)
H_6(x) = 64x^6-480x^4+720x^2-120
(8)
H_7(x) = 128x^7-1344x^5+3360x^3-1680x
(9)
H_8(x) = 256x^8-3584x^6+13440x^4-13440x^2+1680
(10)
H_9(x) = 512x^9-9216x^7+48384x^5-80640x^3+30240x
(11)
H_(10)(x) = 1024x^(10)-23040x^8+161280x^6-403200x^4+302400x^2-30240.
(12)
When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 2; -2, 4; -12, 8; 12, -48, 16; 120, -160, 32; ... (OEIS A059343).
The values H_n(0) may be called Hermite numbers.
The Hermite polynomials are a Sheffer sequence with
g(t) = e^(t^2/4)
(13)
f(t) = 1/2t
(14)
(Roman 1984, p. 30), giving the exponential generating function
exp(2xt-t^2)=sum_(n=0)^infty(H_n(x)t^n)/(n!).
(15)
Using a Taylor series shows that
H_n(x) = [(partial/(partialt))^nexp(2xt-t^2)]_(t=0)
(16)
= [e^(x^2)(partial/(partialt))^ne^(-(x-t)^2)]_(t=0).
(17)
Since partialf(x-t)/partialt=-partialf(x-t)/partialx,
H_n(x) = (-1)^ne^(x^2)[(partial/(partialx))^ne^(-(x-t)^2)]_(t=0)
(18)
= (-1)^ne^(x^2)(d^n)/(dx^n)e^(-x^2).
(19)
Now define operators
O^~_1 = -e^(x^2)d/(dx)e^(-x^2)
(20)
O^~_2 = e^(x^2/2)(x-d/(dx))e^(-x^2/2).
(21)
It follows that
O^~_1f = -e^(x^2)d/(dx)[fe^(-x^2)]
(22)
= 2xf-(df)/(dx)
(23)
O^~_2f = e^(x^2/2)(x-d/(dx))[fe^(-x^2/2)]
(24)
= xf+xf-(df)/(dx)
(25)
= 2xf-(df)/(dx),
(26)
so
O^~_1=O^~_2,
(27)
and
-e^(x^2)d/(dx)e^(-x^2)=e^(x^2/2)(x-d/(dx))e^(-x^2/2)
(28)
(Arfken 1985, p. 720), which means the following definitions are equivalent