Polygamma function


In mathematics, the polygamma function of order is a meromorphic function on the complex numbers defined as the th derivative of the logarithm of the gamma function:
Thus
holds where is the digamma function and is the gamma function. They are holomorphic on. At all the nonpositive integers these polygamma functions have a pole of order. The function is sometimes called the trigamma function.

Integral representation

When and, the polygamma function equals
This expresses the polygamma function as the Laplace transform of. It follows from Bernstein's theorem on monotone functions that, for and real and non-negative, is a completely monotone function.
Setting in the above formula does not give an integral representation of the digamma function. The digamma function has an integral representation, due to Gauss, which is similar to the case above but which has an extra term.

Recurrence relation

It satisfies the recurrence relation
which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:
and
for all. Like the log-gamma function, the polygamma functions can be generalized from the domain Natural number| uniquely to positive real numbers only due to their recurrence relation and one given function-value, say, except in the case where the additional condition of strict monotonicity on is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on is demanded additionally. The case must be treated differently because is not normalizable at infinity.

Reflection relation

where is alternately an odd or even polynomial of degree with integer coefficients and leading coefficient. They obey the recursion equation

Multiplication theorem

The multiplication theorem gives
and
for the digamma function.

Series representation

The polygamma function has the series representation
which holds for and any complex not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as
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,
This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:
Now, the natural logarithm of the gamma function is easily representable:
Finally, we arrive at a summation representation for the polygamma function:
Where is the Kronecker delta.
Also the Lerch transcendent
can be denoted in terms of polygamma function

Taylor series

The Taylor series at is
and
which converges for. Here, 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.

Asymptotic expansion

These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:
and
where we have chosen, i.e. the Bernoulli numbers of the second kind.

Inequalities

The hyperbolic cotangent satisfies the inequality
and this implies that the function
is non-negative for all and. It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that
is completely monotone. The convexity inequality implies that
is non-negative for all and, so a similar Laplace transformation argument yields the complete monotonicity of
Therefore, for all and,