HERMITE EQUATION AND POLYNOMIALS

HERMITE Equation is another special equation like LEGENDRE Differential Equation .  
It is used in the theory of linear harmonic oscillator in quantum mechanics. It is another special form of power series.
The Differential Equation of the form
           

     y" - 2xy' + 2py = 0 ..........(1)

where p is a constant , is called HERMITE DIFFERENTIAL EQUATION.

      Since -2x and 2p are analytic , x= 0 is an ordinary point of equation (1) and has a power series solution  valid for all x .
                    ∞
Let      y =   Σ   cₙ xⁿ           ........(2)
                  n= 0
be a solution of equation (1) .
   Substituting for y, y' ,y" from equation (2) in equation (1) , we obtain
     ∞                            ∞                   ∞
     Σ n(n-1)cₙxⁿ⁻² - 2 Σ ncₙxⁿ + 2pΣ cₙxⁿ = 0
   n=2                        n=1                n=0
       ∞                            ∞
As   Σ n(n-1) cₙ xⁿ⁻² = Σ (n+2)*(n+1)cₙ₊₂xⁿ
     n= 2                       n=0

the above equation can be written as
      ∞
      Σ {(n+2)(n+1)cₙ₊₂ - 2(n-p)cₙ}xⁿ = 0 ........(3)
    n=0

Since equation (3)  holds for all x, the co_efficients of powers of x vanish individually .

Hence (n+2)(n+1)cₙ₊₂ - 2(n-p)cₙ = 0 , n≥0

i.e       cₙ₊₂ = -2(p-n) cₙ / (n+2)(n+1)

Setting n=0,1,2 ..... etc .., we have
             c₂= - 2p c₀/2
             c₃= - 2(p-1) c₁ /3!
             c₄= - 2(p-2) c₂ /4*3 = 2² p(p-2) c₀/4!
     c₅= - 2(p-3) c₃ /5*4 = 2² (p-1)(p-3) c₁/5!  ,etc.

 Substituting these values in (1) , we obtain the power series solution of (1) as

 y = c₀[1- 2p x² /2! + 2² p(p-2) x⁴ / 4! - .....]
        + c₁[x- 2(p-1) x³/3! + 2²(p-1)(p-3) x⁵/5! - ..]

or  y = c₀y₁(x) + c₁y₂(x) (say) .......(4)

where y₁(x) and y₂(x) represent the power series in the first and second brackets respectively. Since c₀ and c₁ are arbitary and y₁and y₂ are lineaely independent , equation (4) represents the general solution of equation (1) .

     If p is a non _ negative integer one of the series terminates and is a polynomial while the other remains as an infinite series . If p is even y₁(x) is a polynomial and if p is odd y₂(x) is a polynomial  and these polynomials for p = 0,1,2,3,4 are respectively  1 , x, 1-2x², x- 2x³/3 , 1- 4x²+4x⁴/3 and constant multiples of these are polynomial solution of the Hermit equation . If we choose the constant so as to have the highest power of x in the form 2ⁿxⁿ, the polynomial solution is called the HERMIT POLYNOMIALS and is denoted by Hₙ(x).
    Hence the polynomial is given by

 Hₙ(x) = (2x)ⁿ - n(n-1) (2x)ⁿ⁻²/1! 
            + n(n-1)(n-2)(n-3) (2x)ⁿ⁻⁴ / 2! - .... .....(5)

 See also:

  Hermit polynomial  in another form.

About Scientist:

  Prof Charles Hermite (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.

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