# Hermite Polynomials: What Are the Initial Values?

• cpsinkule
In summary, the conversation discusses the use of Hermite polynomials defined in a recursive manner and the confusion surrounding the standard values for C0 and C1. The author suggests that the standard values may depend on whether the text is a physics book or not, and provides a link to the definition of Hermite polynomials. The conversation also touches upon the difficulty of generating coefficients using a certain formula and the confusion surrounding the notation used.
cpsinkule
I'm currently reading a text which uses Hermite polynomials defined in the recursive manner. The form of the polynomials are such that C0 C1 are the 0th and 1st terms of a taylor series that generate the remaining coefficients. The author then says the standard value of C1 and C0 are used, but he fails to mention what the standard is?!? Can anyone tell me what the initial values are? I found somewhere that the convention is 2n but I want to make sure.

It's Shankar's QM book. I'm having another difficulty as well. The recursion relation is
Cn+2=Cn(2n+1-2ε)/(n+1)(n+2) where we added the constraint ε=(2n+1)/2. How can I generate any coefficients with this formula? For any N, the numerator is 0 because we had to choose ε that way in order for the series to terminate.

It's a confusing notation. It would be better to say that there must be some ##n=N## such that ##\epsilon = (2N+1)/2## and ##C_{N+2}## = 0. Then the ##C_n## with ##n\leq N## can be nonzero and determined by the recursion relation in terms of ##C_0## or ##C_1##. You should try this out for some particular value like ##N=5##.

Thanks! I realized my mistake. I was thinking the ε was determined by the n in each term of the sum. I see now that the ε you choose for the nth hermite polynomial is the same for every term in the sum so that numerator in the recursion relationship is only 0 for the nth coefficient and up! It is a very confusing notation. He should have a different index for the specific ε than the dummy index in the sum. That's why I got confused.

## What are Hermite polynomials?

Hermite polynomials are a type of orthogonal polynomials that are commonly used in mathematics and physics. They are named after the French mathematician Charles Hermite and are defined by the recurrence relation Hn(x) = 2xHn-1(x) - 2(n-1)Hn-2(x) for n ≥ 1, with H0(x) = 1 and H1(x) = 2x.

## What are the initial values of Hermite polynomials?

The initial values of Hermite polynomials are H0(x) = 1 and H1(x) = 2x.

## What is the significance of the initial values of Hermite polynomials?

The initial values of Hermite polynomials are important because they serve as the starting point for the recurrence relation that defines the polynomials. They also play a crucial role in the derivation of other properties and formulas related to Hermite polynomials.

## How are the initial values of Hermite polynomials derived?

The initial values of Hermite polynomials are derived from the definition of the polynomials and the recurrence relation. By setting n = 0 and n = 1 in the recurrence relation, we can obtain the initial values H0(x) = 1 and H1(x) = 2x.

## What are some applications of Hermite polynomials?

Hermite polynomials have various applications in mathematics and physics. They are commonly used in solving differential equations, probability distributions, and quantum mechanics problems. They also have applications in signal processing, image analysis, and approximation theory.

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