SciPy

scipy.special.sh_jacobi

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

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 \((1 - x)^{p - q}x^{q - 1}\).