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# scipy.special.eval_gegenbauer

scipy.special.eval_gegenbauer(n, alpha, x, out=None) = <ufunc 'eval_gegenbauer'>

  Evaluate Gegenbauer polynomial at a point.

  The Gegenbauer polynomials can be defined via the Gauss hypergeometric function \({}_2F_1\) as

  \[C_n^{(\alpha)} = \frac{(2\alpha)_n}{\Gamma(n + 1)} {}_2F_1(-n, 2\alpha + n; \alpha + 1/2; (1 - z)/2).\]

  When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.46 in [AS] (or equivalently [DLMF]) for details.

  Parameters:
    n array_like

      Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function.

    alpha array_like

      Parameter

    x array_like

      Points at which to evaluate the Gegenbauer polynomial

    out ndarray, optional

      Optional output array for the function values

  Returns:
    C scalar or ndarray

      Values of the Gegenbauer polynomial

  See also

  `roots_gegenbauer`

    roots and quadrature weights of Gegenbauer polynomials

  `gegenbauer`

    Gegenbauer polynomial object

  `hyp2f1`

    Gauss hypergeometric function

  References

  [AS]

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

  [DLMF]

  NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.5.E9