Polygamma function



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


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

Thus


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

holds where $ψ^{(1)}$ is the digamma function and $ψ^{(2)}$ is the gamma function. They are holomorphic on $$\mathbb{C} \backslash\mathbb{Z}_{\le0}$$. At all the nonpositive integers these polygamma functions have a pole of order $ψ^{(3)}$. The function $(m + 1)$ is sometimes called the trigamma function.

Integral representation
When $ψ(z)$ and $Γ(z)$, the polygamma function equals


 * $$\begin{align}

\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\\ &= (-1)^{m+1}m!\zeta(m+1,z) \end{align}$$

where $$\zeta(s,q)$$ is the Hurwitz zeta function.

This expresses the polygamma function as the Laplace transform of $m + 1$. It follows from Bernstein's theorem on monotone functions that, for $ψ^{(1)}(z)$ and $ln Γ(z)$ real and non-negative, $ψ^{(0)}(z)$ is a completely monotone function.

Setting $ψ^{(1)}(z)$ 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 $ψ^{(2)}(z)$ case above but which has an extra term $ψ^{(3)}(z)$.

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 $$n \in \mathbb{N}$$, where $$\gamma$$ is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain $ψ^{(4)}(z)$ uniquely to positive real numbers only due to their recurrence relation and one given function-value, say $m > 0$, except in the case $Re z > 0$ where the additional condition of strict monotonicity on $$\mathbb{R}^{+}$$ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on $$\mathbb{R}^{+}$$ is demanded additionally. The case $(−1)^{m+1} t^{m}⁄1 − e^{−t}$ must be treated differently because $m > 0$ is not normalizable at infinity (the sum of the reciprocals doesn't converge).

Reflection relation

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

where $x$ is alternately an odd or even polynomial of degree $(−1)^{m+1} ψ^{(m)}(x)$ with integer coefficients and leading coefficient $m = 0$. They obey the recursion equation


 * $$\begin{align} P_0(x) &= x \\ P_{m+1}(x) &= - \left( (m+1)xP_m(x)+\left(1-x^2\right)P'_m(x)\right).\end{align}$$

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\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


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

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


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

This relation can for example be used to compute the special values

\psi^{(2n-1)}\left(\frac14\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|+2(2n)!\beta(2n)\right); $$

\psi^{(2n-1)}\left(\frac34\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|-2(2n)!\beta(2n)\right); $$

\psi^{(2n)}\left(\frac14\right) = -2^{2n-1}\left(\pi^{2n+1}|E_{2n}|+2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right); $$

\psi^{(2n)}\left(\frac34\right) = 2^{2n-1}\left(\pi^{2n+1}|E_{2n}|-2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right). $$

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,


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

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


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

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


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

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


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

Where $e^{−t}⁄t$ is the Kronecker delta.

Also the Lerch transcendent
 * $$\Phi(-1, m+1, z) = \sum_{k=0}^\infty \frac{(-1)^k}{(z+k)^{m+1}}$$

can be denoted in terms of polygamma function


 * $$\Phi(-1, m+1, z)=\frac1{(-2)^{m+1}m!}\left(\psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right)$$

Taylor series
The Taylor series at $$\mathbb{N}$$ 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 $ψ^{(m)}(1)$. 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:


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

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

where we have chosen $m = 0$, i.e. the Bernoulli numbers of the second kind.

Inequalities
The hyperbolic cotangent satisfies the inequality
 * $$\frac{t}{2}\operatorname{coth}\frac{t}{2} \ge 1,$$

and this implies that the function
 * $$\frac{t^m}{1 - e^{-t}} - \left(t^{m-1} + \frac{t^m}{2}\right)$$

is non-negative for all $m = 0$ and $ψ^{(0)}$. It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that
 * $$(-1)^{m+1}\psi^{(m)}(x) - \left(\frac{(m-1)!}{x^m} + \frac{m!}{2x^{m+1}}\right)$$

is completely monotone. The convexity inequality $P_{m}$ implies that
 * $$\left(t^{m-1} + t^m\right) - \frac{t^m}{1 - e^{-t}}$$

is non-negative for all $|m − 1|$ and $(−1)^{m}⌈2^{m − 1}⌉$, so a similar Laplace transformation argument yields the complete monotonicity of
 * $$\left(\frac{(m-1)!}{x^m} + \frac{m!}{x^{m+1}}\right) - (-1)^{m+1}\psi^{(m)}(x).$$

Therefore, for all $m > 0$ and $δ_{n0}$,
 * $$\frac{(m-1)!}{x^m} + \frac{m!}{2x^{m+1}} \le (-1)^{m+1}\psi^{(m)}(x) \le \frac{(m-1)!}{x^m} + \frac{m!}{x^{m+1}}.$$

Since both bounds are strictly positive for $$x>0$$, we have: This can be seen in the first plot above.
 * $$\ln\Gamma(x)$$ is strictly convex.
 * For $$m=0$$, the digamma function, $$\psi(x)=\psi^{(0)}(x)$$, is strictly monotonic increasing and strictly concave.
 * For $$m$$ odd, the polygamma functions, $$\psi^{(1)},\psi^{(3)},\psi^{(5)},\ldots$$, are strictly positive, strictly monotonic decreasing and strictly convex.
 * For $$m$$ even the polygamma functions, $$\psi^{(2)},\psi^{(4)},\psi^{(6)},\ldots$$, are strictly negative, strictly monotonic increasing and strictly concave.

Trigamma bounds and asymptote
For the case of the trigamma function ($$m=1$$) the final inequality formula above for $$x>0$$, can be rewritten as:

\frac{x+\frac12}{x^2} \le \psi^{(1)}(x)\le \frac{x+1}{x^2} $$ so that for $$x\gg1$$: $$\psi^{(1)}(x)\approx\frac1x$$.