Hermite polynomials definition

An alternative definition involves the use of a generating function. This method is especially convenient for the evaluation of certain integrals of the Hermite polynomials and can be applied to other polynomials as well. For the Hermite polynomials the generating function can be written as... 

The variable s is a dummy variable in the sense that it does not enter die final result. Thus, if the exponential function in Eq. (94) is expanded in a power series in s, the coefficients of successive powers of s are just the Hermite polynomials divided by u . It is not too difficult to show that Eqs. (93) and (94) are equivalent definitions of the Hermite polynomials. 

For the present purpose, instead of developing the theory of the polynomials // ( ), called the Hermite polynomials, from the relation between successive coefficients given in Equation 11-7, it is more convenient to introduce them by means of another definition ... 

We shall show later that this leads to the same functions as Equation 11-7. A third definition involves the use of a generating function, a method which is useful in many calculations and which is also applicable to other functions. The generating function for the Hermite polynomials is... 

This identity in the auxiliary variable s means that the function hjyg property that, if it is expanded in a power series in 8, the coefficients of successive powers of s are just the Hermite polynomials H ( ), multiplied by 1/n . To show the equivalence of the two definitions 11-13 and 11-14, we differentiate S n times with respect to s and then let s tend to zero, using first one and then the other expression for S the terms with v < n vanish on differentiation, and those with v > n vanish for s —> 0, leaving only the term with v = n ... 

Hamiltonian operator, 140 Heat capacity, definition of, 40 Heisenberg uncertainty principle, 124,148 Hermite polynomials, 79 Hermitian operators, 140 Heterogeneous logarithms, 26 Homogeneous logarithms, 26... 

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