Documentation

# hermiteH

Hermite polynomials

## Syntax

``hermiteH(n,x)``

## Description

example

````hermiteH(n,x)` represents the `n`th-degree Hermite polynomial at the point `x`.```

## Examples

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Find the first five Hermite polynomials for the variable `x`.

```syms x hermiteH([0 1 2 3 4], x)```
```ans = [ 1, 2*x, 4*x^2 - 2, 8*x^3 - 12*x, 16*x^4 - 48*x^2 + 12]```

Depending on whether the input is numeric or symbolic, `hermiteH` returns numeric or exact symbolic results.

Find the value of the fifth-degree Hermite polynomial at `1/3`. Because the input is numeric, `hermiteH` returns numeric results.

`hermiteH(5,1/3)`
```ans = 34.2058```

Find the same result for exact symbolic input. `hermiteH` returns an exact symbolic result.

`hermiteH(5,sym(1/3))`
```ans = 8312/243```

Plot the first five Hermite polynomials.

```syms x y fplot(hermiteH(0:4,x)) axis([-2 2 -30 30]) grid on ylabel('H_n(x)') legend('H_0(x)', 'H_1(x)', 'H_2(x)', 'H_3(x)', 'H_4(x)', 'Location', 'Best') title('Hermite polynomials')```

## Input Arguments

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Degree of the polynomial, specified as a nonnegative integer, symbolic variable, expression, or function, or as a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

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### Hermite Polynomials

Hermite polynomials are defined by this recursion formula.

`$H\left(0,x\right)=1,\text{ }H\left(1,x\right)=2x,\text{ }H\left(n,x\right)=2xH\left(n-1,x\right)-2\left(n-1\right)H\left(n-2,x\right)$`

Hermite polynomials in MATLAB® satisfy this normalization.

`${\int }_{-\infty }^{\infty }{\left({H}_{n}\left(x\right)\right)}^{2}{\text{e}}^{-{x}^{2}}dx={2}^{n}\sqrt{\pi }n!$`

## Tips

• `hermiteH` returns floating-point results for numeric arguments that are not symbolic objects.

• `hermiteH` acts element-wise on nonscalar inputs.

• At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then `hermiteH` expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

## References

[1] Hochstrasser, U. W. “Orthogonal Polynomials.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.