Monday Math 43

Now, we’ve introduced the digamma function, which allows us to find the derivative of the gamma function. But what about higher order derivatives? Here, we find the polygamma functions.
Just as the digamma function is ,
we can define the trigamma function
.
In terms of Γ and its derivatives,
.
So the second derivative of Γ(z) is
.
From the series expression we found for ψ0,

we find

And so we see , where ζ(s) is, as always, the Riemann zeta function.

Further derivatives give us the polygamma function of order n:

From repeated derivatives of , we see that, for n≥1,
.
Thus we have the special values:
.

And from our reccurence relation , we can take the nth derivative to get
.

And we can extend the reflection formula via repeated differentiation to get

From the duplication formula

we find, for n≥1

.
Again letting z=1/2, we have, for n≥1,

Thus , , and .

 

The polygamma functions can be used to find the values of the Euler-Mascheroni integrals: . We already found here that , the Euler-Mascheroni constant. Similarly, we see , where is the nth derivative of the gamma function. We see that

Which, using what we know of these values, gives

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