DiscretePolynomialEnsembles

DiscretePolynomialEnsemble <: PolynomialEnsemble

Abstract type for discrete polynomial ensembles.

Interface

Subtypes must implement:

• ensemble[n](x): evaluate basis polynomial of degree n at x
• weight(ensemble, x): weight function at x
• LinearAlgebra.norm_sqr(ensemble[n]): squared norm of basis element n
• fraction_leading_coefficients(ensemble, n): ratio of leading coefficients of degrees n and n-1

Optionally implement transform(ensemble, x) for coordinate transformations (default: identity).

Kernel(ensemble, n)

A rank-n kernel for ensemble constructed from the first n basis elements, using the Christoffel-Darboux formula.

weight(ensemble, x)

Weight function of ensemble at point x.

BesselJ(; θ)

Bessel J ensemble with parameter θ > 0. Construct the Bessel kernel via Kernel(ensemble).

Charlier(; a)

Charlier polynomial ensemble with parameter a > 0.

DiscreteLegendre(; N)

Discrete Legendre polynomial ensemble on {0, 1, ..., N}.

Hahn(; α, β, M)

Hahn polynomial ensemble with parameters α > -1, β > -1, and integer M.

Krawtchouk(; K, p)

Krawtchouk polynomial ensemble with parameters K (number of trials) and 0 < p < 1 (success probability).

Lanczos(w, domain; ν = identity, simplify = identity)

Orthonormal polynomial ensemble via Lanczos algorithm with weight function w on domain. If a transformation ν is specified, the inner product used for orthonormalization is $⟨f, g⟩ = ∑_{x ∈ 𝒟} w(x) f(ν(x)) g(ν(x))$ and the polynomials will be over ν(x).

LanczosMonic(w, domain; ν = identity, simplify = identity)

Monic polynomial ensemble via Lanczos algorithm with weight function w on domain. If a transformation ν is specified, the inner product used for orthogonalization is $⟨f, g⟩ = ∑_{x ∈ 𝒟} w(x) f(ν(x)) g(ν(x))$ and the polynomials will be over ν(x).

Meixner(; K, q)

Meixner polynomial ensemble with parameters K > 0 and 0 < q < 1.

dJdν(ν, z, precision = 1.0e-14, max_steps = 10^6; prec)

Compute the derivative of the Bessel function J_ν(z) with respect to ν.

digamma_over_gamma(n)

Compute digamma(n)/gamma(n), handling the case when n is a non-positive integer.

fraction_leading_coefficients(ensemble, n)

Ratio of leading coefficients κₙ₋₁/κₙ for basis polynomials of degrees n and n-1.

grad_pFq(pfq_val, a, b, z, precision = 1.0e-14, max_steps = 10^6; prec)

Returns the gradient of the generalized hypergeometric function wrt to the input arguments:

\[ _pF_q(a_1,...,a_p;b_1,...,b_q;z)\]

Where:

\[ \frac{\partial }{\partial a_1} = \sum_{k=1}^{\infty}{ \frac {\left(1 + \sum_{m=0}^{k-1}\frac{1}{m+a_1}\right) * \left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}} - {}_pF_q(a_1,...,a_p;b_1,...,b_q;z)\]

\[ \frac{\partial }{\partial b_1} = {}_pF_q(a_1,...,a_p;b_1,...,b_q;z) - \sum_{k=1}^{\infty}{ \frac {\left(1 + \sum_{m=0}^{k-1}\frac{1}{m+b_1}\right) * \left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}}\]

\[ \frac{\partial }{\partial z} = \frac{\prod_{j=1}^{p}(a_j)}{\prod_{j=1}^{q} (b_j)}\ {}_pF_q(a_1+1,...,a_p+1;b_1+1,...,b_q+1;z)\]

Noting the the recurrence relation for the digamma function: $\psi(x + 1) = \psi(x) + \frac{1}{x}$, the gradients for the function with respect to a and b then simplify to:

\[ \frac{\partial }{\partial a_1} = \sum_{k=1}^{\infty}{ \frac {\left(1 + \sum_{m=0}^{k-1}\frac{1}{m+a_1}\right) * \left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}} - {}_pF_q(a_1,...,a_p;b_1,...,b_q;z)\]

\[ \frac{\partial }{\partial b_1} = {}_pF_q(a_1,...,a_p;b_1,...,b_q;z) - \sum_{k=1}^{\infty}{ \frac {\left(1 + \sum_{m=0}^{k-1}\frac{1}{m+b_1}\right) * \left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}}\]

transform(ensemble, x)

Coordinate transformation applied before polynomial evaluation. Default: identity.

norm_sqr(ensemble[n])

Squared L² norm of basis element n with respect to the ensemble weight.