Multiply a Hermite series by an independent variable in Python

To multiply the Hermite series by x, where x is the independent variable, use the polynomial.hermite.hermmulx() method in Python NumPy. The method returns an array representing the result of the multiplication. The parameter c is a 1-D array of Hermite series coefficients ordered from low to high.

Syntax

numpy.polynomial.hermite.hermmulx(c)

Parameters

c: A 1-D array of Hermite series coefficients ordered from low to high degree.

Example

Let's create a simple example to demonstrate the multiplication of a Hermite series by the independent variable ?

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

# Create an array of Hermite series coefficients
c = np.array([1, 2, 3])

# Display the array
print("Our Array...")
print(c)

# Check the Dimensions
print("\nDimensions of our Array...")
print(c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...")
print(c.dtype)

# Get the Shape
print("\nShape of our Array object...")
print(c.shape)

# Multiply the Hermite series by x
result = H.hermmulx(c)
print("\nResult after multiplication by x...")
print(result)

Our Array...
[1 2 3]

Dimensions of our Array...
1

Datatype of our Array object...
int64

Shape of our Array object...
(3,)

Result after multiplication by x...
[2.  6.5 1.  1.5]

How It Works

The hermmulx() function multiplies each term of the Hermite series by the independent variable x. For a Hermite series represented as c[0]*H?(x) + c[1]*H?(x) + c[2]*H?(x), multiplication by x transforms it according to Hermite polynomial recurrence relations.

Multiple Coefficient Example

Here's another example with different coefficients ?

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

# Create arrays with different coefficients
coeffs1 = np.array([5, 0, 2])
coeffs2 = np.array([1, 1, 1, 1])

print("Original coefficients [5, 0, 2]:")
print("Result after x multiplication:", H.hermmulx(coeffs1))

print("\nOriginal coefficients [1, 1, 1, 1]:")
print("Result after x multiplication:", H.hermmulx(coeffs2))

Original coefficients [5, 0, 2]:
Result after x multiplication: [0.  12.5  0.   1. ]

Original coefficients [1, 1, 1, 1]:
Result after x multiplication: [1.  2.5 1.5 0.5 0.5]

Conclusion

The hermmulx() method efficiently multiplies Hermite series by the independent variable x using polynomial recurrence relations. It's essential for manipulating Hermite polynomials in numerical computations and mathematical modeling.