Multiply a Hermite_e series by an independent variable in Python

To multiply a Hermite_e series by the independent variable x, use the hermemulx() method from NumPy's polynomial module. This method takes a 1-D array of Hermite_e series coefficients and returns the result of multiplying the series by x.

Syntax

numpy.polynomial.hermite_e.hermemulx(c)

Parameters

• c − 1-D array of Hermite_e series coefficients ordered from low to high degree

Return Value

Returns an array representing the Hermite_e series multiplied by x.

Example

Let's create a Hermite_e series and multiply it by the independent variable ?

import numpy as np
from numpy.polynomial import hermite_e as H

# Create coefficient array for Hermite_e series
c = np.array([1, 2, 3])

# Display the array
print("Coefficients:", c)
print("Dimensions:", c.ndim)
print("Datatype:", c.dtype)
print("Shape:", c.shape)

# Multiply the Hermite_e series by x
result = H.hermemulx(c)
print("\nResult after multiplying by x:", result)

Coefficients: [1 2 3]
Dimensions: 1
Datatype: int64
Shape: (3,)

Result after multiplying by x: [2. 7. 2. 3.]

How It Works

The hermemulx() method implements the mathematical operation of multiplying a Hermite_e polynomial by its independent variable. For a Hermite_e series with coefficients [1, 2, 3], representing the polynomial 1*He_0(x) + 2*He_1(x) + 3*He_2(x), multiplying by x transforms the coefficients according to Hermite_e polynomial properties.

Multiple Coefficients Example

Here's an example with different coefficient arrays ?

import numpy as np
from numpy.polynomial import hermite_e as H

# Test with different coefficient arrays
coeff_arrays = [
    np.array([1]),
    np.array([1, 1]),
    np.array([1, 2, 3, 4])
]

for i, coeffs in enumerate(coeff_arrays):
    result = H.hermemulx(coeffs)
    print(f"Input {i+1}: {coeffs}")
    print(f"Output {i+1}: {result}")
    print()

Input 1: [1]
Output 1: [1. 0.]

Input 2: [1 1]
Output 2: [1. 1. 1.]

Input 3: [1 2 3 4]
Output 3: [2. 9. 2. 3. 4.]

Conclusion

The hermemulx() method efficiently multiplies Hermite_e series by the independent variable x. This operation is useful in polynomial mathematics and scientific computing applications involving Hermite_e polynomials.