Polygamma function

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For Barnes's gamma function, see multiple gamma function.

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.

The logarithm of the gamma function and the first few polygamma functions in the complex plane
Complex LogGamma.jpg
Complex Polygamma 0.jpg
Complex Polygamma 1.jpg

\ln\Gamma(z)

\psi^{(0)}(z)

\psi^{(1)}(z)
Complex Polygamma 2.jpg
Complex Polygamma 3.jpg
Complex Polygamma 4.jpg

\psi^{(2)}(z)

\psi^{(3)}(z)

\psi^{(4)}(z)

Integral representation[edit]

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[edit]

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[edit]

(-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[edit]

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[edit]

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[edit]

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[edit]

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.

See also[edit]

References[edit]