Polygamma function

For Barnes's gamma function, see multiple gamma function.
Graphs of the polygamma functions ψ, ψ₁, ψ₂ and ψ₃ of real arguments

In mathematics, the polygamma function of order m is a meromorphic function on $\C$ and defined as the (m+1)-th derivative of the logarithm of the gamma function:

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

Thus

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

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on $\C \setminus -\N_0$. 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.

 $\ln\Gamma(z)$ $\psi^{(0)}(z)$ $\psi^{(1)}(z)$ $\psi^{(2)}(z)$ $\psi^{(3)}(z)$ $\psi^{(4)}(z)$

Integral representation

The polygamma function may be represented as

\begin{align} \psi^{(m)}(z)&= (-1)^{m+1}\int_0^\infty\frac{t^m e^{-zt}} {1-e^{-t}}\ dt\\ &=-\int_0^1\frac{t^{z-1}}{1-t}\ln^mt\ dt \end{align}

which holds for Re z >0 and m > 0. For m = 0 see the digamma function definition.

Recurrence relation

It satisfies the recurrence relation

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

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

$\frac{\psi^{(m)}(n)}{(-1)^{m+1}\,m!} = \zeta(1+m) - \sum_{k=1}^{n-1} \frac{1}{k^{m+1}} = \sum_{k=n}^\infty \frac{1}{k^{m+1}} \qquad m \ge 1$

and

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

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

Reflection relation

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

where $P_m$ is alternatingly an odd resp. even polynomial of degree $|m-1|$ with integer coefficients and leading coefficient $(-1)^m \lceil 2^{m-1}\rceil$. They obey the recursion equation $P_{m+1}(x) = - \left( (m+1)xP_m(x)+(1-x^2)P_m^\prime(x)\right)$ with $P_0(x)=x$.

Multiplication theorem

The multiplication theorem gives

$k^{m+1} \psi^{(m)}(kz) = \sum_{n=0}^{k-1} \psi^{(m)}\left(z+\frac{n}{k}\right)\qquad m \ge 1$

and

$k \psi^{(0)}(kz) = k\log(k) + \sum_{n=0}^{k-1} \psi^{(0)}\left(z+\frac{n}{k}\right)$

for the digamma function.

Series representation

The polygamma function has the series representation

$\psi^{(m)}(z) = (-1)^{m+1}\; m!\; \sum_{k=0}^\infty \frac{1}{(z+k)^{m+1}}$

which holds for 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

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

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,

$\frac{1}{\Gamma(z)} = z \; \mbox{e}^{\gamma z} \; \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) \; \mbox{e}^{-z/n}$. This is a result of the Weierstrass factorization theorem.

Thus, the gamma function may now be defined as:

$\Gamma(z) = \frac{\mbox{e}^{-\gamma z}}{z} \; \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right)^{-1} \; \mbox{e}^{z/n}$

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

$\ln \Gamma(z) = -\gamma z - \ln(z) + \sum_{n=1}^{\infty} \left( \frac{z}{n} - \ln(1 + \frac{z}{n}) \right)$

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

$\psi^{(n)}(z) = \frac{d^{n+1}}{dz^{n+1}}\ln \Gamma(z) = -\gamma \delta_{n0} \; - \; \frac{(-1)^n n!}{z^{n+1}} \; + \; \sum_{k=1}^{\infty} \left(\frac{1}{k} \delta_{n0} \; - \; \frac{(-1)^n n!}{(k+z)^{n+1}}\right)$

Where $\delta_{n0}$ is the Kronecker delta.

Also the alternating series

$\sum_{k=0}^\infty \frac{(-1)^k}{(z+k)^{m+1}}$

can be denoted in term of polygamma function

$\sum_{k=0}^\infty\frac{(-1)^{k}}{(z+k)^{m+1}}=\frac1{(-2)^{m+1}m!}\left[\psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right]$

Taylor series

The Taylor series at z = 1 is

$\psi^{(m)}(z+1)= \sum_{k=0}^\infty (-1)^{m+k+1} \frac {(m+k)!}{k!} \; \zeta (m+k+1)\; z^k \qquad m \ge 1$

and

$\psi^{(0)}(z+1)= -\gamma +\sum_{k=1}^\infty (-1)^{k+1}\zeta (k+1)\;z^k$

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:

$\psi^{(m)}(z) = (-1)^{m+1}\sum_{k=0}^{\infty}\frac{(k+m-1)!}{k!}\frac{B_k}{z^{k+m}} \qquad m \ge 1$

and

$\psi^{(0)}(z) = \ln(z) - \sum_{k=1}^\infty \frac{B_k}{k z^k}$

where we have chosen $B_1 = 1/2$, i.e. the Bernoulli numbers of the second kind.