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:

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

Here

:$psi(z)\; =psi^\{(0)\}(z)\; =\; frac\{Gamma\text{'}(z)\}\{Gamma(z)\}$

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

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.

Recurrence relation

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

Multiplication theorem

The multiplication theorem gives

:$k^\{m\}\; psi^\{(m-1)\}(kz)\; =\; sum\_\{n=0\}^\{k-1\}\; psi^\{(m-1)\}left(z+frac\{n\}\{k\}\; ight)$

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

:$k\; (psi(kz)-log(k))\; =\; sum\_\{n=0\}^\{k-1\}\; psileft(z+frac\{n\}\{k\}\; ight).$

eries 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,

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

Thus, the gamma function may now be defined as:

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

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

$ln\; Gamma(z)\; =\; -gamma\; z\; -\; ln(z)\; +\; sum\_\{n=1\}^\{infty\}\; frac\{z\}\{n\}\; -\; ln(1\; +\; frac\{z\}\{n\})$

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

$psi^\{(n)\}(z)\; =\; frac\{d^\{n+1\{dz^\{n+1ln\; Gamma(z)\; =\; -gamma\; delta\_\{n0\}\; ;\; -\; ;\; frac\{(-1)^n\; n!\}\{z^\{n+1\; ;\; +\; ;\; sum\_\{k=1\}^\{infty\}\; frac\{1\}\{k\}\; delta\_\{n0\}\; ;\; -\; ;\; frac\{(-1)^n\; n!\}\{(1+frac\{z\}\{k\})^\{n+1\}\; ;\; k^\{n+1$

Where $delta\_\{n0\}$ is the Kronecker delta.

"(Aaron Brookner, 2008)"

Taylor 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, &zeta; 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

* Milton Abramowitz and Irene A. Stegun, "Handbook of Mathematical Functions", (1964) Dover Publications, New York. ISBN 978-0-486-61272-0 . See section [http://www.math.sfu.ca/~cbm/aands/page_260.htm &sect;6.4]