applying generating function
The generating function of a function sequence carries information common to the members of the sequence. It may be utilised for deriving various properties, such as recurrence relations, orthogonality properties etc. We take as example
the http://planetmath.org/node/11980generating function of the of Hermite polynomials, and derive from it a recurrence relation and the orthonormality (http://planetmath.org/Orthonormal) formula.
1. First we form the partial derivative with respect to of both of (1):
Here we substitute (1) to the left hand side and rewrite the right hand side, getting
where we can compare the coefficients of :
Thus we have gotten the recurrence relation
Differentiating (1) partially with respect to enables respectively to find a formula expressing the derivative
via the themselves.
2. We copy the equation (1) twice in the forms
multiply these with each other and by and then integrate the obtained equation termwise over :
Thus we can infer that
which implies the orthonormality relation
Cf. Hermite polynomials.
|Title||applying generating function|
|Date of creation||2013-03-22 19:06:58|
|Last modified on||2013-03-22 19:06:58|
|Last modified by||pahio (2872)|