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Hermite polynomials. Hermite’s differential equation. Generating function. Recurrence formulas.



Hermite’s differential equation. The equation

 

1)        y" - 2xy' + 2ny = 0


If n = 0, 1, 2, 3, .... the solution of Hermite’s equation is given by Hermite’s polynomial Hn(x).



Hermite polynomial. The polynomial given by Rodrigue’s formula



ole.gif

 



First few Hermite polynomials


             ole1.gif




Generating function


ole2.gif



Recurrence formulas


ole3.gif




Orthogonality of Hermite polynomials


ole4.gif



Orthogonal series


ole5.gif


where


             ole6.gif




Miscellaneous formulas


             ole7.gif




                                          Source: Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)




References.


1. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)



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