Polygamma function

In mathematics, the polygamma function of order $m$ is a meromorphic function on $ℂ$ and 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 $ℂ \ −ℕ0$ . 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 polygamma function may be represented as

which holds for $Re z > 0$ and $m > 0$ . For $m = 0$ see the digamma function definition.

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 $n \in ℕ$ . Like the log-gamma function, the polygamma functions can be generalized from the domain $ℕ$ 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 monotony on $ℝ +$ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on $ℝ +$ 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).

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