Orthogonal Polynomials & Special Functions

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Abstractions about named special functions and orthogonal polynomial families — Gegenbauer, Faber, Lommel, Zolotarev, and Pidduck polynomials — along with their generating identities, generalized hypergeometric forms, and classification theorems such as Bochner's theorem and the Christoffel–Darboux formula.

14 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bochner's theorem (orthogonal polynomials) — A classification theorem identifying the classical orthogonal-polynomial sequences that are eigenfunctions of a second-order differential operator with polynomial coefficients.
• Christoffel–Darboux formula — An identity collapsing a finite weighted sum of products of orthogonal polynomials into a quotient involving only two consecutive polynomial degrees.
• Cunningham function — A special-function family expressed through the confluent hypergeometric U function and used in higher-order density expansions and diffusion equations.
• Faber polynomials — Polynomials canonically associated with a normalized Laurent series or conformal map, defined by canceling the principal part of its powers.
• Fox–Wright function — A generalized hypergeometric-type function whose series allows affine step sizes in gamma-function parameters.
• Gauss–Lucas theorem — The roots of the derivative of a nonconstant complex polynomial lie in the convex hull of the polynomial's roots.
• Gegenbauer polynomials — An orthogonal-polynomial family on [−1,1] with weight (1−x²)^(alpha−½), generalizing Legendre and Chebyshev polynomials.
• K-function — The special function extending the hyperfactorial to complex arguments through a functional equation involving powers and the gamma function.
• Lambert series — A generating series of the form sum a_n q^n divided by one minus q^n, whose expanded coefficients are divisor sums of the original sequence.
• Lommel polynomial — A polynomial in the reciprocal argument that expresses shifted-order Bessel functions through a two-term basis of neighboring Bessel orders.
• Pidduck polynomials — A named polynomial sequence defined by an exponential generating function involving the ratio of one plus t to one minus t.
• Sombrero function — The radial two-dimensional analogue of sinc, commonly defined as 2J1(πρ)/(πρ), and arising as the Fourier transform of a circular aperture.
• Trigonometric integral — A family of special functions defined by nonelementary integrals involving sine or cosine divided by the integration variable, including the sine and cosine integrals.
• Zolotarev polynomials — Extremal polynomials with prescribed leading coefficients that minimize uniform deviation on an interval, generalizing Chebyshev polynomials in approximation theory.