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Concept
Polygamma function
Summary
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:
Thus
holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on . 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.
Digamma function#Integral representations
When m > 0 and Re z > 0, the polygamma function equals
where is the Hurwitz zeta function.
This expresses the polygamma function as the Laplace transform of (−1)m+1 tm/1 − e−t. It follows from Bernstein's theorem on monotone functions that, for m > 0 and x real and non-negative, (−1)m+1 ψ(m)(x) 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 e−t/t.
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 , where is the Euler–Mascheroni constant. 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).
where Pm is alternately an odd or even polynomial of degree with integer coefficients and leading coefficient (−1)m⌈2m − 1⌉. They obey the recursion equation
The multiplication theorem gives
and
for the digamma function.
Official source
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