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 $\mathbb {C}$ 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

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