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9 Polygamma and Clausen(case An other look at the particular case of the
function
9.1 Polygamma FunctionsThe polygamma functions are defined for
So we are dealling with hypergeometric series :
We'll notice that for However, if I've introduce the polygamma after
their series is to make the link with what interest us here, we need to
know that the usual definition of the polygamma are done using the
famous Gamma function
and we can define In fact, I started from the nice formula
recalling the function Let us start from the well know equality
giving the following formula
Thanks to the never the less famous
equivalence by Stirling extend to the non-integer factorial we have for
by decomposing and sorting the
or
This gives us the logarithmic form of the Gamma function
which is equivalent to the formulation of the
function by the infinite product of Weierstrass for friends. In fact,
by taking into account that From there, we can go in crazy direction (which will keep quiet the rigourous justification because we don't want to overload the page) and easily find the polygamma and digamma functions commonly defined. In fact the first diffirentiation gives
then a second differentiation gives
and so on. Basicly, you've understood it, here we are playing with the Gamma function and it's derivatives.
9.2 The digamma functionWe have seen the definition of the digamma
function above (317).
It is only interesting in the fact that it involves some
From the formula 317,
the constant At the rational points, the digamma function reveals it's intimacy thanks to the theorem on Gauss' digamma, which say that
for We clearly see here the intervention of
9.3
Polygamma of order
Plouffe but in
evidence on his page
numerous relation between the polygamma function of order
9.3.1 Link with the integrals of BBP formulaeEven if those relation were only put to light
later on, they are nothing less than BBP formula with no powers
For this reason, the proof of those formulae are done exactly done the same as in the paragraph dedicated to BBP formulae.
9.3.2 A graphical approachPlouffe approached
the linear relations from an other angle than that of classical
formulae, it's the linear link. In fact, by systematically searching
for relations for the polygamma of order Concerning notation, we'll notice that the
differentiation of the digamma function is the polygamma function of
order 2 : A first example is given with the constants
Here, we need to understand that the circuit
between
As Plouffe said it,
note that if we introduced I also take his commenteries, only common
sense, which indicates for example on the diagram we can not apparently
find relations in The symetry that appears in this scheme (and sometime others) is often due to analytical formula of symetry, here for example, we have
which gives a lot of rational relations at
least for Finaly, a few liasion between rationals
polygamma without Here are the other diagrams build by Plouffe : Order 2
Ordre 3
9.3.3 Analytical translationHere are a few example of relation, where we
see even the dilogarithm appear! We can find the set of relations
discovered by Plouffe on this page
and surf through the sections dedicated to Euler's gamma constant, or Order 2
Order 3
![]()
![]() among others... Proof Here is the proof af a formula, for example : The most well know must be
Proof. The
proof here uses the serie and the integral representation. We can
already note that according to the equivalent integral form (see 322), or in equivalent serie
The right hand side is the definition of
Catalan's constant We can also see all of this more directly by manipulating Euler's serie
by cutting the serie among the even and the odd, i.e.
From here, we have This kind of easy but characteristic is found in all relations of this section. We mess around with the serie, or we go back to the integrals (or a mixture of the two). It's often easier than for the BBP formulae strictly speaking because the cconsidered integral is always between 0 and 1 (in it's usual form)
9.4 Kölbig's
combination
|
![]() |
(343) |
We have in fact straight away
The asymetry of the definition (why not
consider ?) comes from the fact that for
opposite parities, the formulae can only be calculated explicitely by
diverse classical method of integration and analysisp... For example,
we have
![]() |
(345) |
or even
![]() |
(346) |
It's a lovely mathematical problem to
understand why it does not work so well in both cases! It's a similar
problems to the one that concern the odd and even values of Riemann's
function .
A classical result (since there exist some !) is that
![]() |
(347) |
which is a good way to link those two constants !
Clausen's functions are know that they can be
express in functions of known constant uniquely for a little number of
the value of . Which is not very interesting normaly...
But we are going to see that by using the polygamma functions, we can
sometime links them to something else !
Introduction To come to Kölbig's result, we can first of all notce that an integration gives
![]() |
(348) |
by decompostition of the fraction in simpler
elements. The relation is valid for . It's an other way to look at the BBP formulae, without
no interest since in the demonstration, we are going to see for example
that the nature of those constant that intervin are not determined by
the coefficient
,
it being a simple weighting in front of the logarithm. It is said that
the relation is thanks to Jensen and can be found by developing the
logarithm ! Me, I don;t see
where the difficulty was by passing through the decomposition in simple
elements, but I could be wrong ?
Then, the continuity relation by Abel allows to write that for the digamma function,
There we go.
Greater Orders In fact, Kölbig obtained very similar result, but on the differentiation on the digamma function, which we know are the polygamma. Instead of having some logarithm, we have some polylogarithm (logical!) and instead of a logarithm function of the complex exponential, we have polylog of the complex exponential, which we just know are Clausen's function according to 344 ! All of this is very coherant and gives the following Kölbig's relation
![]() |
(350) |
![]() |
(351) |
which is directly proven by the definition of
polylogarithm by summing on :
![]() |
(352) |
We can notice that we obtain the famous
multiplication formula for the polylogarithm by letting
in this equation, it's nice !
I won't go into the details of the nine page long proof in Kölbig's article by the principle is the following: Since the left hand side is real, we taje the real part of the right hand side and we calculate the differentiation of what appears. By considering the general inverse relation, and by using the fact that
![]() |
(353) |
for ,
we finaly obtain that
Theorem 15 of Kölbig
For ,
,
and the even orders, for
,
,
where is the number of Bernoulli's indice
.
Some applications Above (342), we have showed that
![]() |
(356) |
but we can also note that we have
![]() |
(357) |
which allows us to obtain the values for and
from the recursion formulae
![]() |
(358) |
and with some thinking
![]() |
(359) |
but the main interest in those formulae is to
give some values for ,
,
,
in function of well known constants ! For
,
We hence find the relations (325), (331), (332), (335) and (336) given by Plouffe .