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} ight)^m psi(z) = left(frac{d}{dz} ight)^{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.

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]


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