Polygamma function

In mathematics, the polygamma function of order m is a meromorphic function on $\mathbb {C}$ 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 $\mathbb {C} \setminus -\mathbb {N} _{0}$ . 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.

**The logarithm of the gamma function and the first few polygamma functions in the complex plane**

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

$\psi ^{(2)}(z)$	$\psi ^{(3)}(z)$	$\psi ^{(4)}(z)$

Integral representation

The polygamma function may be represented as

{\begin{aligned}\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 t)^{m}\ dt\end{aligned}}

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\geq 1

and

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

for all $n\in \mathbb {N}$ . Like the $\ln \Gamma$ -function, the polygamma functions can be generalized from the domain $\mathbb {N}$ 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 $\mathbb {R} ^{+}$ 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 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}}{dz^{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\geq 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 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)={\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

\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

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

and

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

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

References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 978-0-486-61272-0 . See section §6.4