Polygamma function

In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:

Graphs of the polygamma functions ψ, ψ(1), ψ(2) and ψ(3) of real arguments

Thus

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on \ −0. At all the nonpositive integers these polygamma functions have a pole of order m + 1. 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
Complex LogGamma.jpg
Complex Polygamma 0.jpg
Complex Polygamma 1.jpg
ln Γ(z) ψ(0)(z) ψ(1)(z)
Complex Polygamma 2.jpg
Complex Polygamma 3.jpg
Complex Polygamma 4.jpg
ψ(2)(z) ψ(3)(z) ψ(4)(z)

Integral representationEdit

When m > 0 and Re z > 0, 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 m > 0 and x real and non-negative,   is a completely monotone function.

Setting m = 0 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 m = 0 case above but which has an extra term  .

Recurrence relationEdit

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 n. Like the log-gamma function, the polygamma functions can be generalized from the domain uniquely to positive real numbers only due to their recurrence relation and one given function-value, say ψ(m)(1), except in the case m = 0 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 m = 0 must be treated differently because ψ(0) is not normalizable at infinity (the sum of the reciprocals doesn't converge).

Reflection relationEdit

 

where Pm is alternately an odd or even polynomial of degree |m − 1| with integer coefficients and leading coefficient (−1)m⌈2m − 1. They obey the recursion equation

 

Multiplication theoremEdit

The multiplication theorem gives

 

and

 

for the digamma function.

Series representationEdit

The polygamma function has the series representation

 

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

 

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 δn0 is the Kronecker delta.

Also the Lerch transcendent

 

can be denoted in terms of polygamma function

 

Taylor seriesEdit

The Taylor series at z = 1 is

 

and

 

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.

Asymptotic expansionEdit

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 B1 = 1/2, i.e. the Bernoulli numbers of the second kind.

InequalitiesEdit

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 m ≥ 1 and x > 0,

 

See alsoEdit

ReferencesEdit

  • Abramowitz, Milton; Stegun, Irene A. (1964). "Section 6.4". Handbook of Mathematical Functions. New York: Dover Publications. ISBN 978-0-486-61272-0.