Page 80 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 3: Hermite Polynomials

Notes
( /2) (2n 2 )!x n 2
n
r
= ( 1) r n P n ( )
x
r
2 ( )!(n 2 )!(n r )!
r
n 0
Hence,
2 2
n
dt
P n ( ) t e t H n ( ) .
x
xt
! n 0
Self Assessment
x
12. From recurrence relation II Obtain the value of H 3 ( ). Given that
x
x
H 2 ( ) 4x 2 2; H 1 ( ) 2x
13. Prove that
x
x
H n ( ) 4nx H n 1 ( ) 2n H n ( ) 0
x
14. Prove that
dH 3 ( ) 6H ( )
x
x
dx 2
3.6 Summary
 Hermite differential equation has no finite singular points except x . Therefore
Frobenius method involving a power series solution is obtained.
 There are two independent solutions corresponding to two different values of indicial
power.
 For n a polynomial solution called Hermite polynomial is obtained.

 Hermite polynomials are seen to be generated by a generating function.
 Orthogonal properties of Hermite polynomials are obtained. It helps in expressing any
x
polynomial in terms of H n ( ) .
 Recurrence relations established help in expressing every polynomial as well as its
derivatives in terms of two or three Hermite polynomials.

3.7 Keywords

Boundary Conditions are the behaviour of the solution of the differential equations in the
initial value of the independent variable as well as at the final value of independent variable.

Frobenius Method: At an ordinary point as well as at regular singular point, helps in evaluating
the solution as a power series.
Orthogonality relations of Hermite polynomials are relations involving integrals of two
Hermite polynomials. These relations help us to see that H n ( ) form a complete set.
x
Recurrence Relations are relations between two or three polynomials for all values of n and x.
Rodrigue Formula Expresses H n ( ) in an alternative way than that of finding a solution of
x
differential equations.

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