POLYGAMMA

The polygamma family generalizes the digamma function to higher derivatives of the logarithm of the gamma function. It is useful in advanced approximation theory, statistical inference, and special-function series expansions.

For nonnegative integer order n, it is:

\psi^{(n)}(x)=\frac{d^{n+1}}{dx^{n+1}}\ln\Gamma(x)

This wrapper evaluates \psi^{(n)}(x) for real x and integer derivative order.

Excel Usage

=POLYGAMMA(n, x)

n (int, required): Nonnegative derivative order.
x (float, required): Real input value.

Returns (float): Polygamma value of the specified order at the input.

Example 1: Zeroth-order polygamma equals digamma

Inputs:

n	x
0	3

Excel formula:

=POLYGAMMA(0, 3)

Expected output:

0.922784

Example 2: First-order polygamma at moderate input

Inputs:

n	x
1	3

Excel formula:

=POLYGAMMA(1, 3)

Expected output:

0.394934

Example 3: Second-order polygamma at larger input

Inputs:

n	x
2	10

Excel formula:

=POLYGAMMA(2, 10)

Expected output:

-0.0110498

Example 4: First-order polygamma at fractional input

Inputs:

n	x
1	2.5

Excel formula:

=POLYGAMMA(1, 2.5)

Expected output:

0.490358

Python Code

from scipy.special import polygamma as scipy_polygamma

def polygamma(n, x):
    """
    Compute the n-th derivative of the digamma function.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.polygamma.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        n (int): Nonnegative derivative order.
        x (float): Real input value.

    Returns:
        float: Polygamma value of the specified order at the input.
    """
    try:
        if n < 0:
            return "Error: n must be nonnegative"
        return float(scipy_polygamma(n, x))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

n *

Nonnegative derivative order.

x *

Real input value.