applying generating function

The generating function of a functionMathworldPlanetmath 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

e2zt-t2=n=0Hn(z)n!tn, (1)

the function of the of Hermite polynomialsDlmfDlmfDlmfMathworldPlanetmath, and derive from it a recurrence relation and the orthonormality ( formula.

1.  First we form the partial derivativeMathworldPlanetmath with respect to t 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 tn:

2zHnn!-2Hn-1(n-1)!=Hn+1n!  (n=1, 2,)

Thus we have gotten the recurrence relation

Hn+1(z)= 2zHn(z)-2nHn-1(z)  (n=1, 2,). (2)

Differentiating (1) partially with respect to z enables respectively to find a formula expressing the derivative Hn(z) via the themselves.

2.  We copy the equation (1) twice in the forms


multiply these with each other and by e-x2 and then integrate the obtained equation termwise over :

m=0n=0(-e-x2Hm(x)Hn(x)𝑑x)tmunm!n!= -e-x2e2xt-t2e2xu-u2𝑑x
= -e2x(t+u)-t2-u2-x2𝑑x
= -e-[(t+u)2-2(t+u)x+x2]+2tu𝑑x
= e2tu-e-[x-(t+u)]2𝑑x
= e2tu-e-y2𝑑y
= e2tuπ
= ȷ=0π2jtjujj!
= m=0n=0(πn!2nδmn)tmun

Thus we can infer that


which implies the orthonormality relation

-e-x2Hm(x)Hn(x)𝑑x= 2mm!δmnπ. (3)

Cf. Hermite polynomials.

Title applying generating function
Canonical name ApplyingGeneratingFunction
Date of creation 2013-03-22 19:06:58
Last modified on 2013-03-22 19:06:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Example
Classification msc 33B99
Classification msc 33C45
Classification msc 26C05
Classification msc 26A42
Related topic AreaUnderGaussianCurve