Graphs of the polygamma functions ψ, ψ(1), ψ(2) and ψ(3) of real arguments

In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers ${\displaystyle \mathbb {C} }$ defined as the (m + 1)th derivative of the logarithm of the gamma function:

${\displaystyle \psi ^{(m)}(z):={\frac {\mathrm {d} ^{m)){\mathrm {d} z^{m))}\psi (z)={\frac {\mathrm {d} ^{m+1)){\mathrm {d} z^{m+1))}\ln \Gamma (z).}$

Thus

${\displaystyle \psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)))}$

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on ${\displaystyle \mathbb {C} \backslash -\mathbb {N} _{0))$. At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function.

 ln Γ(z) ψ(0)(z) ψ(1)(z) ψ(2)(z) ψ(3)(z) ψ(4)(z)

## Integral representation

When m > 0 and Re z > 0, the polygamma function equals

{\displaystyle {\begin{aligned}\psi ^{(m)}(z)&=(-1)^{m+1}\int _{0}^{\infty }{\frac {t^{m}e^{-zt)){1-e^{-t))}\,\mathrm {d} t\\&=-\int _{0}^{1}{\frac {t^{z-1)){1-t))(\ln t)^{m}\,\mathrm {d} t.\end{aligned))}

This expresses the polygamma function as the Laplace transform of (−1)m+1 tm/1 − et. It follows from Bernstein's theorem on monotone functions that, for m > 0 and x real and non-negative, (−1)m+1 ψ(m)(x) is a completely monotone function.

Setting m = 0 in the above formula does not give an integral representation of the digamma function. The digamma function has an integral representation, due to Gauss, which is similar to the m = 0 case above but which has an extra term et/t.

## Recurrence relation

It satisfies the recurrence relation

${\displaystyle \psi ^{(m)}(z+1)=\psi ^{(m)}(z)+{\frac {(-1)^{m}\,m!}{z^{m+1))))$

which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:

${\displaystyle {\frac {\psi ^{(m)}(n)}{(-1)^{m+1}\,m!))=\zeta (1+m)-\sum _{k=1}^{n-1}{\frac {1}{k^{m+1))}=\sum _{k=n}^{\infty }{\frac {1}{k^{m+1))}\qquad m\geq 1}$

and

${\displaystyle \psi ^{(0)}(n)=-\gamma \ +\sum _{k=1}^{n-1}{\frac {1}{k))}$

for all ${\displaystyle n\in \mathbb {N} }$. Like the log-gamma function, the polygamma functions can be generalized from the domain ${\displaystyle \mathbb {N} }$ uniquely to positive real numbers only due to their recurrence relation and one given function-value, say ψ(m)(1), except in the case m = 0 where the additional condition of strict monotonicity on ${\displaystyle \mathbb {R} ^{+))$ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on ${\displaystyle \mathbb {R} ^{+))$ is demanded additionally. The case m = 0 must be treated differently because ψ(0) is not normalizable at infinity (the sum of the reciprocals doesn't converge).

## Reflection relation

${\displaystyle (-1)^{m}\psi ^{(m)}(1-z)-\psi ^{(m)}(z)=\pi {\frac {\mathrm {d} ^{m)){\mathrm {d} z^{m))}\cot {\pi z}=\pi ^{m+1}{\frac {P_{m}(\cos {\pi z})}{\sin ^{m+1}(\pi z)))}$

where Pm is alternately an odd or even polynomial of degree |m − 1| with integer coefficients and leading coefficient (−1)m⌈2m − 1. They obey the recursion equation

{\displaystyle {\begin{aligned}P_{0}(x)&=x\\P_{m+1}(x)&=-\left((m+1)xP_{m}(x)+\left(1-x^{2}\right)P'_{m}(x)\right).\end{aligned))}

## Multiplication theorem

The multiplication theorem gives

${\displaystyle k^{m+1}\psi ^{(m)}(kz)=\sum _{n=0}^{k-1}\psi ^{(m)}\left(z+{\frac {n}{k))\right)\qquad m\geq 1}$

and

${\displaystyle k\psi ^{(0)}(kz)=k\ln {k}+\sum _{n=0}^{k-1}\psi ^{(0)}\left(z+{\frac {n}{k))\right)}$

for the digamma function.

## Series representation

The polygamma function has the series representation

${\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}\,m!\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{m+1))))$

which holds for integer values of m > 0 and any complex z not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as

${\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}\,m!\,\zeta (m+1,z).}$

Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

One more series may be permitted for the polygamma functions. As given by Schlömilch,

${\displaystyle {\frac {1}{\Gamma (z)))=ze^{\gamma z}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n))\right)e^{-{\frac {z}{n))}.}$

This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:

${\displaystyle \Gamma (z)={\frac {e^{-\gamma z)){z))\prod _{n=1}^{\infty }\left(1+{\frac {z}{n))\right)^{-1}e^{\frac {z}{n)).}$

Now, the natural logarithm of the gamma function is easily representable:

${\displaystyle \ln \Gamma (z)=-\gamma z-\ln(z)+\sum _{k=1}^{\infty }\left({\frac {z}{k))-\ln \left(1+{\frac {z}{k))\right)\right).}$

Finally, we arrive at a summation representation for the polygamma function:

${\displaystyle \psi ^{(n)}(z)={\frac {\mathrm {d} ^{n+1)){\mathrm {d} z^{n+1))}\ln \Gamma (z)=-\gamma \delta _{n0}-{\frac {(-1)^{n}n!}{z^{n+1))}+\sum _{k=1}^{\infty }\left({\frac {1}{k))\delta _{n0}-{\frac {(-1)^{n}n!}{(k+z)^{n+1))}\right)}$

Where δn0 is the Kronecker delta.

Also the Lerch transcendent

${\displaystyle \Phi (-1,m+1,z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k)){(z+k)^{m+1))))$

can be denoted in terms of polygamma function

${\displaystyle \Phi (-1,m+1,z)={\frac {1}{(-2)^{m+1}m!))\left(\psi ^{(m)}\left({\frac {z}{2))\right)-\psi ^{(m)}\left({\frac {z+1}{2))\right)\right)}$

## Taylor series

The Taylor series at z = 1 is

${\displaystyle \psi ^{(m)}(z+1)=\sum _{k=0}^{\infty }(-1)^{m+k+1}{\frac {(m+k)!}{k!))\zeta (m+k+1)z^{k}\qquad m\geq 1}$

and

${\displaystyle \psi ^{(0)}(z+1)=-\gamma +\sum _{k=1}^{\infty }(-1)^{k+1}\zeta (k+1)z^{k))$

which converges for |z| < 1. Here, ζ is the Riemann zeta function. This series is easily derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

## Asymptotic expansion

These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:

${\displaystyle \psi ^{(m)}(z)\sim (-1)^{m+1}\sum _{k=0}^{\infty }{\frac {(k+m-1)!}{k!)){\frac {B_{k)){z^{k+m))}\qquad m\geq 1}$

and

${\displaystyle \psi ^{(0)}(z)\sim \ln(z)-\sum _{k=1}^{\infty }{\frac {B_{k)){kz^{k))))$

where we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.

## Inequalities

The hyperbolic cotangent satisfies the inequality

${\displaystyle {\frac {t}{2))\operatorname {coth} {\frac {t}{2))\geq 1,}$

and this implies that the function

${\displaystyle {\frac {t^{m)){1-e^{-t))}-\left(t^{m-1}+{\frac {t^{m)){2))\right)}$

is non-negative for all m ≥ 1 and t ≥ 0. It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that

${\displaystyle (-1)^{m+1}\psi ^{(m)}(x)-\left({\frac {(m-1)!}{x^{m))}+{\frac {m!}{2x^{m+1))}\right)}$

is completely monotone. The convexity inequality et ≥ 1 + t implies that

${\displaystyle \left(t^{m-1}+t^{m}\right)-{\frac {t^{m)){1-e^{-t))))$

is non-negative for all m ≥ 1 and t ≥ 0, so a similar Laplace transformation argument yields the complete monotonicity of

${\displaystyle \left({\frac {(m-1)!}{x^{m))}+{\frac {m!}{x^{m+1))}\right)-(-1)^{m+1}\psi ^{(m)}(x).}$

Therefore, for all m ≥ 1 and x > 0,

${\displaystyle {\frac {(m-1)!}{x^{m))}+{\frac {m!}{2x^{m+1))}\leq (-1)^{m+1}\psi ^{(m)}(x)\leq {\frac {(m-1)!}{x^{m))}+{\frac {m!}{x^{m+1))}.}$