1%feature("docstring") OT::LaguerreFactory
2"Laguerre specific orthonormal univariate polynomial family.
3
4For the :class:`~openturns.Gamma` distribution.
5
6Available constructors:
7    LaguerreFactory(*k=1.0, ot.LaguerreFactory.ANALYSIS*)
8
9Parameters
10----------
11k : float
12    If `parameters_set == ot.JacobiFactory.PROBABILITY`: default shape
13    parameter :math:`k > 0` of the :class:`~openturns.Gamma` distribution.
14
15    If `parameters_set == ot.JacobiFactory.ANALYSIS`: alternative shape
16    parameter :math:`k_a = k - 1 > -1` of the :class:`~openturns.Gamma`
17    distribution.
18parameters_set : int, optional
19    Integer telling which parameters set is used for defining the distribution
20    (amongst `ot.LaguerreFactory.ANALYSIS, ot.LaguerreFactory.PROBABILITY`).
21
22Notes
23-----
24Any sequence of orthogonal polynomials has a recurrence formula relating any
25three consecutive polynomials as follows:
26
27.. math::
28
29    P_{i + 1} = (a_i x + b_i) P_i + c_i P_{i - 1}, \quad 1 < i
30
31The recurrence coefficients for the Laguerre polynomials come analytically and
32read:
33
34.. math::
35
36    \begin{array}{rcl}
37        a_i & = & \omega_i \\
38        b_i & = & - (2 i + k_a + 1) \omega_i \\
39        c_i & = & - \sqrt{(i + k_a) i} \omega_i
40    \end{array}, \quad 1 < i
41
42where :math:`k_a` is the alternative shape parameter of the
43:class:`~openturns.Gamma` distribution, and:
44
45.. math::
46
47    \omega_i = \frac{1}{\sqrt{(i + 1) (i + k_a + 1)}} , \quad 1 < i
48
49See also
50--------
51StandardDistributionPolynomialFactory
52
53Examples
54--------
55>>> import openturns as ot
56>>> polynomial_factory = ot.LaguerreFactory()
57>>> for i in range(3):
58...     print(polynomial_factory.build(i))
591
60-1 + X
611 - 2 * X + 0.5 * X^2"
62
63// ---------------------------------------------------------------------
64
65%feature("docstring") OT::LaguerreFactory::getK
66"Accessor to the alternative shape parameter :math:`k_a`.
67
68Of the :class:`~openturns.Gamma` distribution.
69
70Returns
71-------
72k_a : float
73    Alternative shape parameter :math:`k_a = k - 1` of the
74    :class:`~openturns.Gamma` distribution."
75