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Special Functions

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13 domain-specific abstractions whose origin domain is Special Functions.

  • Bessel Function — A parameterized special-function family solving Bessel's equation and furnishing radial modes for cylindrical separation problems.
  • Cunningham function — A special-function family expressed through the confluent hypergeometric U function and used in higher-order density expansions and diffusion equations.
  • Digamma function — The logarithmic derivative ψ(z)=Γ′(z)/Γ(z) of the gamma function, extending shifted harmonic-number relations to complex arguments.
  • Fox–Wright function — A generalized hypergeometric-type function whose series allows affine step sizes in gamma-function parameters.
  • Gegenbauer polynomials — An orthogonal-polynomial family on [−1,1] with weight (1−x²)^(alpha−½), generalizing Legendre and Chebyshev polynomials.
  • Incomplete polylogarithm — A special function obtained by truncating the integral representation of the polylogarithm at a nonzero lower limit, also related to incomplete Fermi–Dirac and Bose–Einstein integrals.
  • Inverse tangent integral — The special function Ti₂(x)=∫₀ˣ arctan(t)/t dt, equivalently an odd dilogarithmic combination with a characteristic alternating odd-power series.
  • K-function — The special function extending the hyperfactorial to complex arguments through a functional equation involving powers and the gamma function.
  • Lommel polynomial — A polynomial in the reciprocal argument that expresses shifted-order Bessel functions through a two-term basis of neighboring Bessel orders.
  • Meijer G-function — Define a highly general special function by a Mellin–Barnes contour integral whose gamma-factor parameters subsume many hypergeometric and classical functions.
  • Mott polynomials — A polynomial sequence defined by an exponential generating function involving the Catalan-series expression (sqrt(1−t²)−1)/t, introduced in connection with electron theory.
  • Pidduck polynomials — A named polynomial sequence defined by an exponential generating function involving the ratio of one plus t to one minus t.
  • Quarter period — The complete elliptic-integral quantities K(m) and iK′(m) that generate the period lattice of Jacobi elliptic functions and locate their characteristic quarter-cycle values.