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scipy.special.

# jacobi

scipy.special.jacobi(n, alpha, beta, monic=False)

  Jacobi polynomial.

  Defined to be the solution of

  \[\begin{split}\begin{aligned} (1 - x^2)\frac{d^2}{dx^2} P_n^{(\alpha, \beta)}(x) &+ \left(\beta - \alpha - (\alpha + \beta + 2)x\right) \frac{d}{dx} P_n^{(\alpha, \beta)}(x) \\ &+ n(n + \alpha + \beta + 1) P_n^{(\alpha, \beta)}(x) = 0 \end{aligned}\end{split}\]

  for \(\alpha, \beta > -1\); \(P_n^{(\alpha, \beta)}\) is a polynomial of degree \(n\).

  Parameters:
    n int

      Degree of the polynomial.

    alpha float

      Parameter, must be greater than -1.

    beta float

      Parameter, must be greater than -1.

    monic bool, optional

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

  Returns:
    P orthopoly1d

      Jacobi polynomial.

  Notes

  For fixed \(\alpha, \beta\), the polynomials \(P_n^{(\alpha, \beta)}\) are orthogonal over \([-1, 1]\) with weight function \((1 - x)^\alpha(1 + x)^\beta\).

  References

  [AS]

  Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

  Examples

  The Jacobi polynomials satisfy the recurrence relation:

  \[P_n^{(\alpha, \beta-1)}(x) - P_n^{(\alpha-1, \beta)}(x) = P_{n-1}^{(\alpha, \beta)}(x)\]

  This can be verified, for example, for \(\alpha = \beta = 2\) and \(n = 1\) over the interval \([-1, 1]\):

      >>> import numpy as np
      >>> from scipy.special import jacobi
      >>> x = np.arange(-1.0, 1.0, 0.01)
      >>> np.allclose(jacobi(0, 2, 2)(x),
      ...             jacobi(1, 2, 1)(x) - jacobi(1, 1, 2)(x))
      True

  Plot of the Jacobi polynomial \(P_5^{(\alpha, -0.5)}\) for different values of \(\alpha\):

      >>> import matplotlib.pyplot as plt
      >>> x = np.arange(-1.0, 1.0, 0.01)
      >>> fig, ax = plt.subplots()
      >>> ax.set_ylim(-2.0, 2.0)
      >>> ax.set_title(r'Jacobi polynomials $P_5^{(\alpha, -0.5)}$')
      >>> for alpha in np.arange(0, 4, 1):
      ...     ax.plot(x, jacobi(5, alpha, -0.5)(x), label=rf'$\alpha={alpha}$')
      >>> plt.legend(loc='best')
      >>> plt.show()