Hypergeometric functions \(\,_2F_1\) and \(\,_1F_2\) (and related functions)

Special functions

• Gauss Hypergeometric Function \(\,_2F_1\)
  • Gauss Hypergeometric Function, \({}_2F_1(a,b;c;x)\)
    • ctx.hyperg_2f1()
  • Regularized Hypergeometric Function, \({}_2\widetilde{F}_1(a,b;c;x)\)
    • ctx.hyperg_2f1r()
• Chebyshev, Gegenbauer and Jacobi polynomials
  • Chebyshev polynomial (or function) of the first kind, \(T_n(x)\)
    • ctx.chebyshev_t()
  • Chebyshev polynomial of the second kind, \(U_n(x)\)
    • ctx.chebyshev_u()
  • Chebyshev polynomials of the third kind, \(V_n(x)\)
    • math53.chebyshev_v()
  • Chebyshev polynomials of the fourth kind, \(W_n(x)\)
    • math53.chebyshev_w()
  • Gegenbauer (ultraspherical) polynomial, \(C_{n}^{\alpha}(x)\)
    • ctx.gegenbauer_c()
  • Jacobi polynomials, \(P_{n}^{(a, b)}\)
    • ctx.jacobi_p()
  • Zernike radial polynomials \(R_n^m(r)\)
    • math53.zernike_r()
• Legendre polynomials and related
  • Legendre polynomial (or function) of the first kind, \(P_n(x)\)
    • ctx.legendre_p()
  • Associated Legendre function of the first kind, \(P^m_l(x)\)
    • ctx.legendre_plm()
  • Legendre function of the second kind, \(Q_l(x)\)
    • ctx.legendre_q()
  • Associated Legendre function of the second kind, \(Q^m_l(x)\)
    • ctx.legendre_qlm()
  • Spherical harmonics, \(Y_n^m(\theta, \phi)\)
    • ctx.spherical_harmonic()
  • Toroidal harmonics \(P^m_{l-1/2}(x)\)
    • math53.toroidal_plm()
  • Toroidal harmonics \(Q^m_{l-1/2}(x)\)
    • math53.toroidal_qlm()
  • Olver’s associated Legendre function \(Q^m_{l-1/2}(x)\)
    • math53.olver_qlm()
• Incomplete beta functions
  • Non-normalized lower incomplete beta function, \(B_{\mathrm{lower}}(a,b;x)\)
    • ctx.beta_lower()
  • Normalized incomplete beta function, \(I_{x}(a,b)\)
    • ctx.ibeta()
  • Derivative of the incomplete beta function
    • ctx.real_ibeta_prime()
  • Real non-normalized upper incomplete beta function, \(B_{\mathrm{upper}}(a,b;x)\)
    • ctx.beta_upper()
  • Real normalized complementory incomplete beta function, \(I_{1-x}(a,b)\)
    • ctx.ibetac()
  • Inverse of the real normalised incomplete beta function
    • ctx.real_ibeta_inv()
  • Inverse of the real normalised complementary incomplete beta function
    • ctx.real_ibetac_inv()
  • Inverse (on parameter \(a\)) of the real normalised incomplete beta function
    • ctx.real_ibeta_inva()
  • Inverse (on parameter \(a\)) of the real normalised complementary incomplete beta function
    • ctx.real_ibetac_inva()
  • Inverse (on parameter \(b\)) of the real normalised incomplete beta function
    • ctx.real_ibeta_invb()
  • Inverse (on parameter \(b\)) of the real normalised complementary incomplete beta function
    • ctx.real_ibetac_invb()
• Hypergeometric function \({}_1F_2\)
  • Non-regularized hypergeometric function \({}_1F_2\)
    • ctxflint.hyperg_1f2()
  • Regularized hypergeometric function \({}_1\widetilde{F}_2\)
    • ctxflint.hyperg_1f2r()
• Scorer functions
  • Scorer function \(\mathrm{Gi}(x)\)
    • math53.scorer_gi()
  • Scorer function \(\mathrm{Hi}(x)\)
    • math53.scorer_hi()
  • First derivative of the Scorer function \(\mathrm{Gi}(x)\), \(\mathrm{Gi}'(x)\)
    • math53.scorer_gi_prime()
  • First derivative of the Scorer function \(\mathrm{Hi}(x)\), \(\mathrm{Hi}'(x)\)
    • math53.scorer_hi_prime()
• Struve functions
  • Struve function \(\mathbf{H}_n(x)\)
    • math53.struve_h()
  • Struve function \(\mathbf{L}_{\nu}(x)\)
    • math53.struveL()
  • Struve function \(\mathbf{K}_{\nu}(x)\)
    • math53.struveK()
  • Struve function \(\mathbf{M}_{\nu}(x)\)
    • math53.struveM()
• Anger, Weber and Lommel functions
  • Anger function \(\mathbf{J}_{\nu}(x)\)
    • CtxFlint.AngerJ()
  • Weber function \(\mathbf{E}_{\nu}(x)\)
    • CtxFlint.WeberE()
  • Lommel function \(s_{\mu,\nu}(x) = s^{(1)}_{\mu,\nu}(x)\)
    • CtxFlint.LommelS1()
  • Lommel function \(S_{\mu,\nu}(x) = s^{(2)}_{\mu,\nu}(x)\)
    • CtxFlint.LommelS2()