Definitions

# Polygamma function

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

$psi^\left\{\left(m\right)\right\}\left(z\right) = left\left(frac\left\{d\right\}\left\{dz\right\}right\right)^m psi\left(z\right) = left\left(frac\left\{d\right\}\left\{dz\right\}right\right)^\left\{m+1\right\} lnGamma\left(z\right).$

Here

$psi\left(z\right) =psi^\left\{\left(0\right)\right\}\left(z\right) = frac\left\{Gamma\text{'}\left(z\right)\right\}\left\{Gamma\left(z\right)\right\}$

is the digamma function and $Gamma\left(z\right)$ is the gamma function. The function $psi^\left\{\left(1\right)\right\}\left(z\right)$ is sometimes called the trigamma function.

The logarithm of the gamma function and the first few polygamma functions in the complex plane
$lnGamma\left(z\right)$ $psi^\left\{\left(0\right)\right\}\left(z\right)$ $psi^\left\{\left(1\right)\right\}\left(z\right)$ $psi^\left\{\left(2\right)\right\}\left(z\right)$ $psi^\left\{\left(3\right)\right\}\left(z\right)$ $psi^\left\{\left(4\right)\right\}\left(z\right)$

## Integral representation

The polygamma function may be represented as

$psi^\left\{\left(m\right)\right\}\left(z\right)= \left(-1\right)^\left\{\left(m+1\right)\right\}int_0^infty$
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.

## Recurrence relation

It has the recurrence relation
$psi^\left\{\left(m\right)\right\}\left(z+1\right)= psi^\left\{\left(m\right)\right\}\left(z\right) + \left(-1\right)^m; m!; z^\left\{-\left(m+1\right)\right\}.$

## Multiplication theorem

The multiplication theorem gives

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

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

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

## Series representation

The polygamma function has the series representation

$psi^\left\{\left(m\right)\right\}\left(z\right) = \left(-1\right)^\left\{m+1\right\}; m!; sum_\left\{k=0\right\}^infty$
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

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

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,

$1 / Gamma\left(z\right) = z ; mbox\left\{e\right\}^\left\{gamma z\right\} ; prod_\left\{n=1\right\}^\left\{infty\right\} left\left(1 + frac\left\{z\right\}\left\{n\right\}right\right) ; mbox\left\{e\right\}^\left\{-z/n\right\}$. This is a result of the Weierstrass factorization theorem.

Thus, the gamma function may now be defined as:

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

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

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

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

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

Where $delta_\left\{n0\right\}$ is the Kronecker delta.

(Aaron Brookner, 2008)

## Taylor series

The Taylor series at z = 1 is

$psi^\left\{\left(m\right)\right\}\left(z+1\right)= sum_\left\{k=0\right\}^infty$
(-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 series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

## References

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