Polygamma function

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In mathematics, the polygamma function of order m is defined as the (m + 1)th derivative of the logarithm of the gamma function:

\psi^{(m)}(z) = \left(\frac{d}{dz}\right)^m \psi(z) = \left(\frac{d}{dz}\right)^{m+1} \ln\Gamma(z)

Here

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

is the digamma function and Γ(z) is the gamma function. 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
lnΓ(z) ψ(0)(z) ψ(1)(z) ψ(2)(z) ψ(3)(z) ψ(4)(z)


Contents

[edit] Integral representation

The polygamma function may be represented as

\psi^{(m)}(z)= (-1)^{(m+1)}\int_0^\infty \frac{t^m e^{-zt}} {1-e^{-t}} dt

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

[edit] Recurrence relation

It has the recurrence relation

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

[edit] Multiplication theorem

The multiplication theorem gives

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

for m > 1, and, for m = 0, one has the digamma function:

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

[edit] 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.

[edit] Taylor's series

The Taylor series at z = 1 is

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

which converges for |z| < 1. Here, ζ is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

[edit] References