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scipy.special.sh_jacobi

scipy.special.sh_jacobi(n, p, q, monic=False)

Shifted Jacobi polynomial.

Defined by

$$G_n^{(p, q)}(x) = \binom{2n + p - 1}{n}^{-1}P_n^{(p - q, q - 1)}(2x - 1),$$

where $P_n^{(\cdot, \cdot)}$ is the nth Jacobi polynomial.

Parameters:

n : int

Degree of the polynomial.

p : float

Parameter, must have $p > q - 1$.

q : float

Parameter, must be greater than 0.

monic : bool, optional

If True, scale the leading coefficient to be 1. Default is False.

Returns:

G : orthopoly1d

Shifted Jacobi polynomial.

Notes

For fixed $p, q$, the polynomials $G_n^{(p, q)}$ are orthogonal over $[0, 1]$ with weight function $(1 - x)^{p - q}x^{q - 1}$.