Lerch’s phi and related

The Lerch transcendent generalizes various other functions:

The polygamma function (see polygamma()) is given by

\[\psi^{(n)}(z) = (-1)^{n+1} n! \: \Phi(1, n+1, z).\]

The polylogarithm (see polylog()) is given by

\[\text{Li}_s(z) = z \Phi(z, s, 1)).\]

The Legendre chi function (see legendre_chi()) is given by

\[\chi _{n}(z)=2^{-n}z\Phi (z^{2},n,1/2).\]

The Hurwitz zeta function (see hurwitz()) is given by

\[\zeta(s,a) = \Phi(1, s, a).\]

The Riemann zeta function (see zeta()) is given by

\[\zeta (s)=\Phi (1,s,1).\]

The Dirichlet eta function (see dirichlet_eta()) is given by

\[\eta (s)=\Phi (-1,s,1).\]

Various identities include:

\[\Phi (z,s,a)=z^{n}\Phi (z,s,a+n)+\sum _{k=0}^{n-1}{\frac {z^{k}}{(k+a)^{s}}}\]

and

\[\Phi (z,s-1,a)=\left(a+z{\frac {\partial }{\partial z}}\right)\Phi (z,s,a)\]

and

\[\Phi (z,s+1,a)=-\,{\frac {1}{s}}{\frac {\partial }{\partial a}}\Phi (z,s,a).\]

Implementation notes

The current Amath implementation is restricted to real arguments \(z \leq 1, s \geq -1, a \geq 0\).

The current ARB implementation is restricted to calls to the polylogarithm or the Hurwitz zeta function, i.e. it expects \(s=0\) for arbitrary \(z\) and \(a\), or \(z=1\) for arbitrary \(s\) and \(a\), or \(a\) a nonnegative integer for arbitrary \(z\) and \(s\), and otherwise returns NAN.

• Lerch’s transcendent and Lerch’s zeta
  • Lerch’s transcendent, \(\Phi(x,s,a)\)
    • ctx.lerch_phi()
  • Lerch’s zeta
    • ctxflint.lerch_zeta()
• Polygamma and related functions
  • Polygamma function, \(\psi^{(n)}(x)\)
    • ctx.polygamma()
  • TriGamma function, \(\psi'(x)\)
    • ctx.trigamma()
  • DiGamma function \(\psi(x)\)
    • ctx.digamma()
    • ctx.psi()
  • Harmonic number function, \(H_x\)
    • math53.harmonic()
• Polylogarithm and related functions
  • Polylogarithm, \(\mathrm{Li}_s(z)\)
    • ctx.polylog()
  • Trilogarithm Function, \(\mathrm{Li}_3(z)\)
    • math53.trilog()
  • Dilogarithm Function, \(\mathrm{Li}_2(z)\)
    • ctx.dilog()
  • Generalized Clausen sine function
    • ctxflint.clausen_sin()
  • Generalized Clausen cosine function
    • ctxflint.clausen_cos()
  • Classical Clausen function, \(\mathrm{Cl}_2(x)\)
    • math53.clausen2()
  • Bose-Einstein integrals, \(G_s(x)\)
    • math53.bose_einstein()
  • Fermi-Dirac integrals, \(F_s(x)\)
    • ctx.fermi_dirac()
  • Legendre’s Chi function, \(\chi_s(x)\)
    • math53.legendre_chi()
  • Generalized inverse tangent integral
    • ctxflint.inverse_tan_integral()
    • math53.tangent_int()
• Hurwitz zeta and related functions
  • Hurwitz zeta function, \(\zeta(s,a)\)
    • ctx.hurwitz_zeta()
  • Generalized harmonic number function, \(H_x^{(r)}\)
    • math53.harmonic2()
  • Bernoulli numbers, \(B_n\)
    • ctx.bernoulli()
  • Bernoulli polynomials, \(B_n(x)\)
    • math53.bernpoly()
  • Euler numbers
    • ctx.eulernum()
  • Euler polynomials, \(E_n(x)\)
    • math53.eulerpoly()
  • Barnes G-function
    • ctxflint.barnes_g()
  • Logarithm of Barnes G function
    • ctx.logbarnes_g()
  • Hyperfactorial
    • ctxflint.hyperfactorial()
  • Superfactorial
    • ctxflint.superfactorial()
• Riemann zeta function, and related functions
  • Riemann zeta function, \(\zeta(s)\)
    • ctx.zeta()
  • Riemann \(\zeta(s)-1\)
    • math53.zetam1()
  • Hardy (or Riemann-Siegel) theta function
    • mathc53.hardy_theta()
  • Hardy (or Riemann-Siegel) Z function
    • mathc53.hardy_z()
  • Riemann (Landau) function \(\xi(s)\)
    • ctxflint.riemann_xi()
  • Dirichlet eta function, \(\eta(s)\)
    • math53.dirichlet_eta()
  • Dirichlet \(\eta(s) - 1\)
    • math53.dirichlet_eta_m1()
  • Dirichlet beta function, \(\beta(s)\)
    • math53.dirichlet_beta()
  • Dirichlet lambda function, \(\lambda(s)\)
    • math53.dirichlet_lambda()
  • Zeros of the Riemann zeta function
    • ctxflint.zeta_zero()