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I've recently been working on a number of integrals related to the loggamma function, so I thought I'd share my results here. I'll have to post as and when I have time, and there will be a fair bit of preliminary work before we get to the final results, but - loosely speaking - the main aim here is to find a closed form for parametric integrals of the form:$$\Gamma i_m(q)=\int_0^qx^m\log\Gamma(x)\,dx$$Where $$0 < q \le 1$$. The special case where $$q=1$$ appears in a number of contemporary papers, but - thus far - I've not encountered the general case for $$q$$.
In the process of evaluating the above integral, we are led to consider a two generalized Clausen Functions, that in turn, can be expressed as finite sums of derivatives of the Hurwitz Zeta function $$\zeta(z,a)$$, where:
$$\zeta(z,a)=\sum_{k=0}^{\infty}\frac{1}{(k+a)^z}$$Definition:
For the purposes of this tutorial, we define the (possibly new? - although I doubt it) generalized Poly-Clausen Functions:$$C_m(n;\theta)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos k\theta}{k^m}=\lim_{\,z\to m}\frac{d^n}{dz^n}\sum_{k=1}^{\infty} \frac{\cos k\theta}{k^z}$$$$S_m(n;\theta)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\sin k\theta}{k^m}=\lim_{\,z\to m}\frac{d^n}{dz^n}\sum_{k=1}^{\infty} \frac{\sin k\theta}{k^z}$$Following the usual convention for CL and SL-type Clausen Functions, where applicable, we will use the following notation, which is entirely consistent with regular Clausen Functions:$$\text{Cl}_{2m}(n;\theta)=\sum_{k=1}^{\infty} \frac{(\log k)^n \sin k \theta}{k^{2m}}$$
$$\text{Cl}_{2m+1}(n;\theta)=\sum_{k=1}^{\infty} \frac {(\log k)^n \cos k \theta}{k^{2m+1}}$$
$$\text{Sl}_{2m}(n;\theta)=\sum_{k=1}^{\infty} \frac{(\log k)^n \cos k \theta}{k^{2m}}$$
$$\text{Sl}_{2m+1}(n;\theta)=\sum_{k=1}^{\infty} \frac {(\log k)^n \sin k \theta}{k^{2m+1}}$$Evaluations of the Generalized Clausen Functions:With regards to evaluating our parametric loggamma integral, $$\Gamma i_m(q)$$, we will have a particular need for the Generalized Clausen functions at $$q=1/p$$, and $$p \in \mathbb{Z}^{+}$$. In other words, we will need to evaluate the functions:$$C_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos \left( \frac{2\pi k}{p}\right)}{k^m}$$
$$S_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\sin \left( \frac{2\pi k}{p}\right)}{k^m}$$Actually, we will only really need the case where $$n=1$$, but there's no harm in evaluating the general case for $$n$$.
As a starting point, we want to split those sums on the RHS into a finite composition of infinite sums. Indeed, we want each sum to split into exactly p-parts, with denominators corresponding to $$(kp+1)$$, $$(kp+2)$$, etc, up to $$(kp +p)$$.
For example, splitting the cosine-type generalized Poly-Clausen Function in such a way, we obtain:
$$C_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{j=1}^p \left( \sum_{k=0}^{\infty}\frac{\log^n(kp+j) \cos \left( \frac{2\pi (kp+j)}{p}\right)}{(kp+j)^m} \right)=$$
$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \left( \sum_{k=0}^{\infty} \frac{\log^n \left[p(k+j/p) \right]}{(k+j/p)^m} \right)$$Next, we expand the logarithmic term via the binomial theorem, like so:
$$\log^n[p(k+j/p)]=\sum_{l=0}^n \binom{n}{l}(\log p)^{n-l}\log^l(k+j/p)$$Which gives us the partial evaluation:
$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{l=0}^n \binom{n}{l}(\log p)^{n-l} \left( \sum_{k=0}^{\infty} \frac{\log^l (k+j/p) }{(k+j/p)^m} \right)$$That final, rightmost (doubly 'nested') sum - within the large brackets - can be expressed in terms of derivatives of the Hurwitz Zeta function.
$$\frac{d^l}{dz^l}\zeta(z,a)=\frac{d^l}{dz^l}\sum_{k=0}^{\infty}\frac{1}{(k+a)^z}=(-1)^l\sum_{k=0}^{\infty} \frac{\log^l (k+a) }{(k+a)^z}$$So, with the understanding that - henceforth - ALL derivatives of the Hurwitz Zeta function included here are derivatives with respect to the first parameter, we may write:$$C_m \left(n;\frac{2\pi}{p}\right)=$$$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{l=0}^n \binom{n}{l}(\log p)^{n-l} (-1)^l\zeta^{(l)}\left(m, \frac{j}{p} \right)$$
Which - with a re-labelling of the summation index - is the result we required:
$$C_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos \left( \frac{2\pi k}{p}\right)}{k^m}=$$
$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{k=0}^n \binom{n}{k}(\log p)^{n-k} (-1)^k\zeta^{(k)}\left(m, \frac{j}{p} \right)$$An equivalent calculation gives us the result for the other Poly-Clausen Function:Results:$$C_m \left(n;\frac{2\pi}{p}\right)=$$$$(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos \left( \frac{2\pi k}{p}\right)}{k^m}=$$$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{k=0}^n \binom{n}{k}(\log p)^{n-k} (-1)^k\zeta^{(k)}\left(m, \frac{j}{p} \right)$$
$$S_m \left(n;\frac{2\pi}{p}\right)=$$$$(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\sin \left( \frac{2\pi k}{p}\right)}{k^m}=$$$$\frac{(-1)^n}{p^m} \sum_{j=1}^{p-1} \sin \left( \frac{2\pi j}{p} \right) \sum_{k=0}^n \binom{n}{k}(\log p)^{n-k} (-1)^k\zeta^{(k)}\left(m, \frac{j}{p} \right)$$
**Notice that, when $$j=p$$ in that last result, we get a vanishing term corresponding to $$\sin 2\pi$$, hence the reduction in the upper limit of the summation index, with $$p$$ being replaced by $$p-1$$.
Comments and/or questions should be posted here:
http://mathhelpboards.com/commentary-threads-53/commentary-generalized-clausen-function-associated-loggamma-7506.html
In the process of evaluating the above integral, we are led to consider a two generalized Clausen Functions, that in turn, can be expressed as finite sums of derivatives of the Hurwitz Zeta function $$\zeta(z,a)$$, where:
$$\zeta(z,a)=\sum_{k=0}^{\infty}\frac{1}{(k+a)^z}$$Definition:
For the purposes of this tutorial, we define the (possibly new? - although I doubt it) generalized Poly-Clausen Functions:$$C_m(n;\theta)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos k\theta}{k^m}=\lim_{\,z\to m}\frac{d^n}{dz^n}\sum_{k=1}^{\infty} \frac{\cos k\theta}{k^z}$$$$S_m(n;\theta)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\sin k\theta}{k^m}=\lim_{\,z\to m}\frac{d^n}{dz^n}\sum_{k=1}^{\infty} \frac{\sin k\theta}{k^z}$$Following the usual convention for CL and SL-type Clausen Functions, where applicable, we will use the following notation, which is entirely consistent with regular Clausen Functions:$$\text{Cl}_{2m}(n;\theta)=\sum_{k=1}^{\infty} \frac{(\log k)^n \sin k \theta}{k^{2m}}$$
$$\text{Cl}_{2m+1}(n;\theta)=\sum_{k=1}^{\infty} \frac {(\log k)^n \cos k \theta}{k^{2m+1}}$$
$$\text{Sl}_{2m}(n;\theta)=\sum_{k=1}^{\infty} \frac{(\log k)^n \cos k \theta}{k^{2m}}$$
$$\text{Sl}_{2m+1}(n;\theta)=\sum_{k=1}^{\infty} \frac {(\log k)^n \sin k \theta}{k^{2m+1}}$$Evaluations of the Generalized Clausen Functions:With regards to evaluating our parametric loggamma integral, $$\Gamma i_m(q)$$, we will have a particular need for the Generalized Clausen functions at $$q=1/p$$, and $$p \in \mathbb{Z}^{+}$$. In other words, we will need to evaluate the functions:$$C_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos \left( \frac{2\pi k}{p}\right)}{k^m}$$
$$S_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\sin \left( \frac{2\pi k}{p}\right)}{k^m}$$Actually, we will only really need the case where $$n=1$$, but there's no harm in evaluating the general case for $$n$$.
As a starting point, we want to split those sums on the RHS into a finite composition of infinite sums. Indeed, we want each sum to split into exactly p-parts, with denominators corresponding to $$(kp+1)$$, $$(kp+2)$$, etc, up to $$(kp +p)$$.
For example, splitting the cosine-type generalized Poly-Clausen Function in such a way, we obtain:
$$C_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{j=1}^p \left( \sum_{k=0}^{\infty}\frac{\log^n(kp+j) \cos \left( \frac{2\pi (kp+j)}{p}\right)}{(kp+j)^m} \right)=$$
$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \left( \sum_{k=0}^{\infty} \frac{\log^n \left[p(k+j/p) \right]}{(k+j/p)^m} \right)$$Next, we expand the logarithmic term via the binomial theorem, like so:
$$\log^n[p(k+j/p)]=\sum_{l=0}^n \binom{n}{l}(\log p)^{n-l}\log^l(k+j/p)$$Which gives us the partial evaluation:
$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{l=0}^n \binom{n}{l}(\log p)^{n-l} \left( \sum_{k=0}^{\infty} \frac{\log^l (k+j/p) }{(k+j/p)^m} \right)$$That final, rightmost (doubly 'nested') sum - within the large brackets - can be expressed in terms of derivatives of the Hurwitz Zeta function.
$$\frac{d^l}{dz^l}\zeta(z,a)=\frac{d^l}{dz^l}\sum_{k=0}^{\infty}\frac{1}{(k+a)^z}=(-1)^l\sum_{k=0}^{\infty} \frac{\log^l (k+a) }{(k+a)^z}$$So, with the understanding that - henceforth - ALL derivatives of the Hurwitz Zeta function included here are derivatives with respect to the first parameter, we may write:$$C_m \left(n;\frac{2\pi}{p}\right)=$$$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{l=0}^n \binom{n}{l}(\log p)^{n-l} (-1)^l\zeta^{(l)}\left(m, \frac{j}{p} \right)$$
Which - with a re-labelling of the summation index - is the result we required:
$$C_m \left(n;\frac{2\pi}{p}\right)=(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos \left( \frac{2\pi k}{p}\right)}{k^m}=$$
$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{k=0}^n \binom{n}{k}(\log p)^{n-k} (-1)^k\zeta^{(k)}\left(m, \frac{j}{p} \right)$$An equivalent calculation gives us the result for the other Poly-Clausen Function:Results:$$C_m \left(n;\frac{2\pi}{p}\right)=$$$$(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\cos \left( \frac{2\pi k}{p}\right)}{k^m}=$$$$\frac{(-1)^n}{p^m} \sum_{j=1}^p \cos \left( \frac{2\pi j}{p} \right) \sum_{k=0}^n \binom{n}{k}(\log p)^{n-k} (-1)^k\zeta^{(k)}\left(m, \frac{j}{p} \right)$$
$$S_m \left(n;\frac{2\pi}{p}\right)=$$$$(-1)^n\sum_{k=1}^{\infty}\frac{(\log k)^n\sin \left( \frac{2\pi k}{p}\right)}{k^m}=$$$$\frac{(-1)^n}{p^m} \sum_{j=1}^{p-1} \sin \left( \frac{2\pi j}{p} \right) \sum_{k=0}^n \binom{n}{k}(\log p)^{n-k} (-1)^k\zeta^{(k)}\left(m, \frac{j}{p} \right)$$
**Notice that, when $$j=p$$ in that last result, we get a vanishing term corresponding to $$\sin 2\pi$$, hence the reduction in the upper limit of the summation index, with $$p$$ being replaced by $$p-1$$.
Comments and/or questions should be posted here:
http://mathhelpboards.com/commentary-threads-53/commentary-generalized-clausen-function-associated-loggamma-7506.html