Definitions

Polygamma_function

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

Here

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

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

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}
psileft(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} 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+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!}{(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, 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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