Polygamma function: Information from Answers.com

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**

$lnΓ(z)$	$ψ (0) (z)$	$ψ (1) (z)$	$ψ (2) (z)$	$ψ (3) (z)$	$ψ (4) (z)$

1 Integral representation
2 Recurrence relation
3 Multiplication theorem
4 Series representation
5 Taylor series
6 See also
7 References

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

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

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

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

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,

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

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

Where $δ n 0$ is the Kronecker delta.

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, 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

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

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	Reflection formula
	Dirichlet beta function
	Multiplication theorem

	Is the inverse of a quadratic function not a function? Read answer...
	What is the function of the SUMIF function? Read answer...
	Does every function have an inverse that is a function? Read answer...

	What is it's function?
	What are the functions of?
	What is are the functions of who?

Polygamma function

Contents