This post is a continuation of Generalization of the Bernoulli polynomials ( in relation to the Index ) , the definition of the Bernoulli polynomial $B_t(x)$ with $|x|<1$ has an extension through $B_t(x+1)=B_t(x)+t x^{t-1}$.

Two equivalent definitions for $B_t(x)$ with $|x|<1$:

$$B_t(x):=-t\zeta(1-t,x)$$ or \begin{align*} B_t(x+1):=&-\frac{2\Gamma(1+t)}{(2\pi)^t}\cos(\frac{\pi t}{2}) \sum\limits_{k=0}^\infty (-1)^k \frac{(2\pi x)^{2k}}{(2k)!}\zeta(t-2k) \\ &-\frac{2\Gamma(1+t)}{(2\pi)^t}\sin(\frac{\pi t}{2}) \sum\limits_{k=0}^\infty (-1)^k \frac{(2\pi x)^{2k+1}}{(2k+1)!}\zeta(t-1-2k) \end{align*} with $-t\in\mathbb{R}\setminus\mathbb{N}$.

With https://www.researchgate.net/publication/238803313_Bernoulli_numbers_and_polynomials_of_arbitrary_complex_indices , page 86, Theorem 5, using equation (11) with the lower limit of $1$ instead of $0$ ($k=1$ instead of $k=0$) the formula for the sum of fractional powers is $$S_x(t):=\sum\limits_{k=1}^x k^t =\frac{B_{t+1}(x+1)-B_{t+1}(1)}{t+1}$$ with $x\in\mathbb{N}_0$ and $t\in\mathbb{R}_0^+$ (general: $t$ can be complex but I don’t need this possibility here).

The right side may be differentiated by $x$ and therefore one can write $$\frac{\partial}{\partial x} S_x(t)=B_t(x+1)$$ On the other hand differentiated by $t$ and the definition with $M_x(t):=\prod\limits_{k=1}^x k^{k^t} $ it's $$\ln M_x(t)=\frac{\partial}{\partial t}S_x(t)=\frac{\partial}{\partial t}\frac{B_{t+1}(x+1)-B_{t+1}(1)}{t+1}$$

Together one gets (by exchanging the derivatives, which is possible here) $$\frac{\partial}{\partial t}B_t(x+1)=\frac{\partial}{\partial x}\ln M_x(t)$$


Perhaps this equation becomes a bit clearer if one looks at $$\frac{\partial}{\partial t}\Delta B_t(x)=\frac{\partial}{\partial x}\Delta \ln M_{x-1}(t)$$ with $\Delta B_t(x):=B_t(x+1)-B_t(x)=tx^{t-1}$ and $\Delta \ln M_x(t):=\ln M_{x+1}(t)-\ln M_x(t)=(x+1)^t\ln(x+1)$.

The problem now is:

I need a formula for $\ln M_x(t)$ or $M_x(t)$, independend of $B_t(x)$ (otherwise it's a trivial identity), where $x$ and $t$ are variable. It could be a series of (more or less known) functions of $x$ (or perhaps $x$ and $t$) which becomes a sum/term for $t\in\mathbb{N}$ - similar to $B_t(x)$.

Alternative: To proof that the two definitions above for $B_t(x)$ are indeed equivalent (a link to the literatur is enough).


The Euler-MacLaurin-formula can perhaps give a formula for $\ln M_x(t)$. Does someone know a link, where this is computed ?


Maybe http://ac.els-cdn.com/S0377042798001927/1-s2.0-S0377042798001927-main.pdf?_tid=36ead884-7132-11e6-ac53-00000aab0f6b&acdnat=1472837296_60501a990f4d37792d48c76ad38c7e4b , page 198, equation (21), can help. (I will see.)

An application example with $\ln M_x(1)$:

The fourier series of $B_t(x)$ is $$\Re(\sum\limits_{k=1}^\infty \frac{e^{i2\pi kx}}{(ik)^t})=\frac{(2\pi)^t}{2\Gamma(1+t)}B_t(x)$$ for $|x|<1$ and $t>0$.

It is known, that $\frac{d}{dx}\ln M_x(1)=-\ln\sqrt{2\pi}+\frac{1}{2}+x+\ln\Gamma(1+x)$.

$$\frac{\partial}{\partial t}B_t(x)|_{t=1}=\frac{d}{dx}\ln M_{x-1}(1)$$ and derivating the fourier series of $B_t(x)$ (above) by $t$ and having regard to $(\ln\Gamma(1+t))'|_{t=1}=1-\gamma$ one gets $$\sum\limits_{k=1}^\infty \frac{\ln k}{k}\sin(2\pi kx)=\frac{\pi}{2}(\ln\frac{\Gamma(x)}{\Gamma(1-x)}-(1-2x)(\gamma +\ln(2\pi)))$$ which can be seen in http://reader.digitale-sammlungen.de/en/fs1/object/display/bsb10525489_00011.html?zoom=1.0 (on the top of page 4) and in http://arxiv.org/pdf/1309.3824.pdf (page 30, formula 65.)

A second application example where I use $\frac{d}{dx}\ln M_x(m+1)|_{x=0}$ with $m\in\mathbb{N}_0$:

Adamchik had computed $$\zeta’(-m)=\frac{B_{m+1}H_m}{m+1}-A_m$$ where $B_n$ are the Bernoulli-numbers, $H_n$ are the harmonic numbers and $A_n$ are the generalized Glaisher-Kinkelin constants. See e.g. http://www.sciencedirect.com/science/article/pii/S0377042798001927 (Article; last page, equation (24)) .

Dissolving the equation (5.4) on page 36 of
for $\ln M_x(k)$, using $\frac{B_{k+1}(x+1+w_2)- B_{k+1}(1+w_2)}{k+1}$ instead of $\sum\limits_{j=1}^x (w_2+j)^k$ and setting $(w_1;w_2):=(1;0)$ results in

\begin{align*} \ln M_x(m)&=H_m\frac{B_{m+1}(x+1)- B_{m+1}(1)}{m+1}+\ln Q_m(x)+ \\ &+\sum\limits_{k=0}^{m-1}\binom{m}{k}(-x)^{m-k}\sum\limits_{v=0}^k \binom{k}{v}x^{k-v}(\ln A_v -\ln Q_v(x)) \end{align*}

The definition of $Q_m(x)$ is (4.2) on page 13, it’s something like a modified Multiple-Gamma-Function. $\frac{d}{dx}\ln M_x(m)$ can be computed by using the differentiation rule (4.4) for the equation above.

Now one gets with $B_t(1)=-t\zeta(1-t)$ and $\frac{d}{dt}B_t(1)|_{t=m}=\frac{d}{dx}\ln M_x(m)|_{x=0}$ the equation chain $$\frac{B_{m+1}(1)}{m+1}+(m+1)\zeta’(-m)= \zeta(-m)+(m+1)\zeta’(-m)=(-t\zeta(1-t))’$$ $$=\frac{d}{dt}B_t(1)|_{t=m+1}=\frac{d}{dx}\ln M_x(m+1)|_{x=0}=H_{m+1}B_{m+1}(1)-(m+1)\ln A_m$$ and this result dissolved for $\zeta’(-m)$ and took into account that $H_{m+1}-\frac{1}{m+1}=H_m$ and $H_m B_{m+1}(1)=H_m B_{m+1}$ for $m\in\mathbb{N}_0$ one gets Adamchik’s result.

Most simple solution for proofing $\displaystyle \frac{\partial}{\partial t}B_t(x+1)=\frac{\partial}{\partial x}\ln M_x(t)$

by using the 2nd development of G Cab with the Hurwitz Zeta function:

$\zeta(a,b):= \sum\limits_{k=0}^\infty (b+k)^{-a}$

$\displaystyle \frac{B_{t+1}(x+1)-B_{t+1}(1)}{t+1}=S_x(t)=\zeta(-t,1)-\zeta(-t,x+1)$ and therefore
$\displaystyle \frac{\partial}{\partial t}S_x(t)=\ln M_x(t)=\sum\limits_{k=0}^\infty (k+1)^t\ln(k+1) - \sum\limits_{k=0}^\infty (k+x+1)^t\ln (k+x+1)$

$\displaystyle \frac{\partial}{\partial x}S_x(t)= B_t(x+1)=-t\zeta(1-t,x+1)\,$ (as mentioned by gammatester, first link above)

\begin{align*} \frac{\partial}{\partial t}B_t(x+1)&= \frac{\partial}{\partial t}\frac{\partial}{\partial x}(\zeta(-t,1)-\zeta(-t,x+1)) \\ &=\frac{\partial}{\partial x}\frac{\partial}{\partial t}(\zeta(-t,1)-\zeta(-t,x+1))=\frac{\partial}{\partial x}\ln M_x(t) \end{align*}


Substituting $B_t(x)$ and $\ln M_x(t)$ by other formulas are leading to non-trivial equations (as shown in the application examples above).

  • $\begingroup$ When it comes to defining things you should be a bit more careful. For example $\sum_{k=1}^x$ carries no meaning as a sum when $x$ is not an integer. I would instead define $M_x$ by the $\partial_t B_{t+1}(x+1) + \ldots$ formula and then note that when $x$ is an integer we have this product formula. Same for $S_x(t)$. $\endgroup$ – Winther Sep 6 '16 at 12:38
  • $\begingroup$ ...I would also explicitly add the generalized $B_t(x)$ formula you assume in the linked question to the question here as it's relevant here. $\endgroup$ – Winther Sep 6 '16 at 12:41
  • $\begingroup$ I know (of course) that this sum has no meaning by itself. :-) But it becomes a meaning by using the Bernoulli-polynomials. I will add the definition, thanks for the hint. The sense of the application examples is to make the text understandable. $\endgroup$ – user90369 Sep 6 '16 at 12:53
  • $\begingroup$ You say that but then you also say “The problem now is I need a formula for $M_x$ independent of $B_t(x)$ (otherwise it's a trivial identity)”. As I read it this seems to be begging for us trying to extract meaning from the product formulation of $M_x$ independent of the definition given by the formula involving $B_t(x)$. I fear this question is ill defined, atleast I don't understand what you are really asking for. $\endgroup$ – Winther Sep 6 '16 at 13:31
  • $\begingroup$ In the ideal case of an answer is a formula for $\ln M_x(t)$ which contains $\ln M_x(m)$, mentioned in the second application example. But any other formula is wellcome, which leads to a nontrivial formula between $B_t(x)$ and $\ln M_x(t)$ - I don't expect only one solution, of course different points of view exist. E.g. G Cab gave me another idea. - Nobody is begged, nobody is forced to understand what I mean. But perhaps someone is interested in such "exotic" problems. And I am glad and thankful about any idea. $\endgroup$ – user90369 Sep 6 '16 at 14:04

This is just a "what if ?" consideration, not an answer, and I just guess that it might be of some help to your scope. So, flanking the analysis you are conducting, you may consider this alternative development for $S_x(t)$.

  • 1st development $$ \begin{gathered} S_x (t) = \sum\limits_{k = 1}^x {k^{\,t} } = \sum\nolimits_{\;k = 1}^{\;x + 1} {k^{\,t} } = \frac{{B_{\,t + 1} (x + 1) - B_{\,t + 1} (1)}} {{t + 1}} = \quad \quad \left( \text{1} \right) \hfill \\ = \sum\nolimits_{\;k = 0}^{\;x} {\left( {k + 1} \right)^{\,t} } = \sum\nolimits_{\;k = 0}^{\;x} {\sum\limits_{0\, \leqslant \,j} {\left( \begin{gathered} t \hfill \\ j \hfill \\ \end{gathered} \right)k^{\,j} } } = \sum\limits_{0\, \leqslant \,j} {\left( \begin{gathered} t \hfill \\ j \hfill \\ \end{gathered} \right)\sum\nolimits_{\;k = 0}^{\;x} {k^{\,j} } } = \hfill \\ = \sum\limits_{0\, \leqslant \,j} {\left( \begin{gathered} t \hfill \\ j \hfill \\ \end{gathered} \right)\left( {\frac{{B_{\,j + 1} (x) - B_{\,j + 1} (0)}} {{j + 1}}} \right)} = \quad \quad \left( 2 \right) \hfill \\ = \sum\nolimits_{\;k = 0}^{\;x} {\sum\limits_{\begin{array}{*{20}c} {0\, \leqslant \,j} \\ {0\, \leqslant \,l\,\left( { \leqslant \,j} \right)} \\ \end{array} } {\left( \begin{gathered} t \hfill \\ j \hfill \\ \end{gathered} \right)\left\{ \begin{gathered} j \\ l \\ \end{gathered} \right\}k^{\,\underline {\,l\,} } } } = \hfill \\ = \sum\limits_{\begin{array}{*{20}c} {0\, \leqslant \,j} \\ {0\, \leqslant \,l\,\left( { \leqslant \,j} \right)} \\ \end{array} } {\left( \begin{gathered} t \hfill \\ j \hfill \\ \end{gathered} \right)\left\{ \begin{gathered} j \\ l \\ \end{gathered} \right\}\frac{{x^{\,\underline {\,l + 1\,} } }} {{l + 1}}} = \sum\limits_{\begin{array}{*{20}c} {0\, \leqslant \,j} \\ {0\, \leqslant \,l\,\left( { \leqslant \,j} \right)} \\ \end{array} } {\frac{{t^{\,\underline {\,j\,} } }} {{j!}}\left\{ \begin{gathered} j \\ l \\ \end{gathered} \right\}\frac{{x^{\,\underline {\,l + 1\,} } }} {{l + 1}}} \quad \quad \left( 3 \right) \hfill \\ \end{gathered} $$ where the symbol $\sum\nolimits_{\;k = 1}^{\;x + 1} {}$ indicates the indefinite sum , computed between the indicated bounds, and the curly backets the Stirling N. of 2nd kind.
    To the purpose of derivating vs. $t$ and $x$ , you may replace the falling factorials $t^{\,\underline {\,j\,} } $ and$x^{\,\underline {\,l + 1\,} } $ with the corresponding Stirling devopment in $t^n$ and $x^m$ or with their expression through the Gamma function.
  • 2nd development
    You can also write $S_x(t)$ in terms of the Hurwitz zeta function $$ \begin{gathered} S_x (t) = \sum\limits_{k = 1}^x {k^{\,t} } = \sum\nolimits_{\;k = 1}^{\;x + 1} {k^{\,t} } = \hfill \\ = \sum\nolimits_{\;k = 1}^{\;\infty } {k^{\,t} } - \sum\nolimits_{\;k = x + 1}^{\;\infty } {k^{\,t} } = \sum\nolimits_{\;k = 0}^{\;\infty } {\left( {k + 1} \right)^{\,t} } - \sum\nolimits_{\;j = 0}^{\;\infty } {\left( {j + x + 1} \right)^{\,t} } = \hfill \\ = \zeta ( - t,1) - \zeta ( - t,x + 1)\quad \quad \left( 4 \right) \hfill \\ \end{gathered} $$
  • Note concerning the handling of sums and products with non-integer bounds
    First let's note that $$ \begin{gathered} S_x (t) = \sum\limits_{k = 1}^x {k^{\,t} } \quad \Rightarrow \hfill \\ \Rightarrow \quad x^{\,t} = S_{x + 1} (t) - S_{x + 1} (t) = \left( {S_{x + 1} (t) + c(x + 1)} \right) - \left( {S_x (t) + c(x)} \right) \hfill \\ \end{gathered} $$ and $$ \begin{gathered} M_x (t) = \prod\limits_{k = 1}^x {k^{\,k^{\,t} } } = \prod\nolimits_{\;k = 1\;}^{\;x + 1} {k^{\,k^{\,t} } } = \prod\nolimits_{\;k = 0\;}^{\;x} {\left( {k + 1} \right)^{\,\left( {k + 1} \right)^{\,t} } } \quad \Rightarrow \hfill \\ \Rightarrow \quad \left( {x + 1} \right)^{\,\left( {x + 1} \right)^{\,t} } = \frac{{M_{x + 1} (t)}} {{M_x (t)}} = \frac{{c(x + 1)M_{x + 1} (t)}} {{c(x)M_x (t)}} \hfill \\ \end{gathered} $$ with $$ c(x)\;:\quad \text{any}\,\text{periodic}\,\text{function}\text{,}\,\text{with}\,\text{period}\,\;1 $$ Then let's take for example the starting base of your development, we get the following two different "definitions" for $B_t(x+1)$ $$ \begin{array}{*{20}c} {S_x (t) = \frac{{B_{\,t + 1} (x + 1) - B_{\,t + 1} (1)}} {{t + 1}}} \hfill & \begin{gathered} \hfill \\ = \hfill \\ \hfill \\ \end{gathered} \hfill & \begin{gathered} = \sum\limits_{k = 1}^x {k^{\,t} } = \sum\nolimits_{\;k = 1}^{\;x + 1} {k^{\,t} } = \hfill \\ = \sum\nolimits_{\;k = 0}^{\;\infty } {\left( {k + 1} \right)^{\,t} } - \sum\nolimits_{\;k = 0}^{\;\infty } {\left( {k + x + 1} \right)^{\,t} } \hfill \\ \end{gathered} \hfill \\ \hline \begin{gathered} \quad \quad \quad \quad \Downarrow \hfill \\ \frac{\partial } {{\partial \,x}}S_x (t) = \hfill \\ = \frac{1} {{t + 1}}\frac{\partial } {{\partial \,x}}B_{\,t + 1} (x + 1) = \hfill \\ = B_{\,t} (x + 1) = \hfill \\ = - t\sum\nolimits_{\;k = 0}^{\;\infty } {\left( {k + x + 1} \right)^{\,t - 1} } \hfill \\ \end{gathered} \hfill & \begin{gathered} | \hfill \\ | \hfill \\ | \hfill \\ | \hfill \\ | \hfill \\ | \hfill \\ | \hfill \\ | \hfill \\ \end{gathered} \hfill & \begin{gathered} \quad \quad \quad \quad \Downarrow \hfill \\ \frac{{B_{\,t + 1} (x + 1)}} {{t + 1}} = f(t + 1) - \sum\nolimits_{\;k = 0}^{\;\infty } {\left( {k + x + 1} \right)^{\,t} } \hfill \\ \quad \quad \quad \quad \Downarrow \hfill \\ \hfill \\ B_{\,t} (x + 1) = \hfill \\ = t\,f(t) - t\sum\nolimits_{\;k = 0}^{\;\infty } {\left( {k + x + 1} \right)^{\,t - 1} } \hfill \\ \end{gathered} \hfill \\ \end{array} $$ where
  • the derivate in $x$ is first taken in extending to real index the known property for integer index, and then by derivating the espression of $S(x)$ as difference of the two sums;
  • $f(t)$ can be any function in $t$, and in particular it could be $B_t(1)$, which in turn can be taken as $t\;\zeta (1 - t)$, as it is in many papers concerning the extension of Bernoulli polynomials.

Thus it is evident that such mathematical entities shall be handled with great care, and specially when taking derivatives.

  • $\begingroup$ Thank you for your efforts! - I need a term of functions for $\ln M_x(t)$ so that we have new insights with combining it with the Bernoulli-polynomials. I think only to develope series doesn't help to proof the equation between $B_t(x+1)$ and $\ln M_x(t)$, but who knows, perhaps it works (I will try). Anyway, you have helped me that I could make my question at least a bit clearer, thanks. $\endgroup$ – user90369 Sep 3 '16 at 14:04
  • $\begingroup$ I have added a comment at the end of my post above. The problem is that I don't get a closed form for the spezial case $t\in\mathbb{N}$. $\endgroup$ – user90369 Sep 6 '16 at 11:54
  • $\begingroup$ @user90369 a) I checked formulas 1 & 2 & 3 with my old CAS for 0 <= integer $t$ and $x$ <= 10 and re-verified the various steps: could not find errors b) added to my answer the derivation in terms of Hurwitz zeta c) cannot follow you in the use of $M_x(t)$ etc., specially because I am missing the final scope: is your aim to find a formulation for the derivative of $S_x(t)$ vs. $x$ and $t$ , or to find the properties of $M_x(t)$, or else ? $\endgroup$ – G Cab Sep 6 '16 at 14:39
  • $\begingroup$ Thank you for your addition! (1) For the proof of $\frac{\partial}{\partial t}B_t(x+1)=\frac{\partial}{\partial x}\ln M_x(t)$ I need an expression for $\ln M_x(t)$ with variable $x$ and $t$. (2) The ideal case of an answer is a formula for $\ln M_x(t)$ which contains $\ln M_x(m)$ mentioned in the second application example. But of course any idea is welcome. Perhaps the term with Hurwitz zeta function will work, I will see. :-) $\endgroup$ – user90369 Sep 6 '16 at 14:55
  • $\begingroup$ Thanks again! You fill with $c(x)$ the space between two natural numbers, that's o.k. . But please define $f(t)$ and $g(t)$ before the first use and it's better to bring the lines in an order ( I cannot see the equations clearly because the lines of the left and right part of the equations are not on the same height), I am a bit confused. :-) $\endgroup$ – user90369 Sep 7 '16 at 14:53

Applying the Euler-Maclaurin Sum Formula

The Euler-Maclaurin Sum Formula can be applied to $k^t$ to get the approximation $$ \sum_{k=1}^nk^t=\zeta(-t)+\frac1{t+1}n^{t+1}+\frac12n^t+\frac{t}{12}n^{t-1}-\frac{t^3-3t^2+2t}{720}n^{t-2}+O\left(n^{t-3}\right) $$ When $t\lt-1$, this describes how the series for $\zeta(-t)$ converges.

Possible Extension to Non-Integral Summation Limits

Consider $$ \begin{align} \lim_{\delta\to0}\frac1\delta\left(\sum_{k=1}^{n+\delta}k^t\color{#C00000}{-\sum_{k=1}^{m+\delta}k^t}\right) &=\lim_{\delta\to0}\frac1\delta\sum_{k=m+1+\delta}^{n+\delta}k^t\\ &=\lim_{\delta\to0}\frac1\delta\sum_{k=m+1}^n(k+\delta)^t\\ &=t\sum_{k=m+1}^nk^{t-1} \end{align} $$ Thus, if we give a meaning to taking a derivative with respect to the upper limit of summation, it would give $$ \frac{\mathrm{d}}{\mathrm{d}n}\sum_{k=1}^nk^t=t\sum_{k=1}^nk^{t-1}\color{#C00000}{+C} $$ where $C$ is related to the behavior near $m=0$.

  • $\begingroup$ Thank you for answering my question of the note! But with this formula I have still the problem to derivate by $n$. $\endgroup$ – user90369 Sep 3 '16 at 16:10
  • $\begingroup$ @user90369: I have added a section about a possible extension that might allow differentiation with respect to a limit of summation. $\endgroup$ – robjohn Sep 3 '16 at 16:38
  • $\begingroup$ An interesting addition, thanks. But to derivate $\sum\limits_{k=1}^n k^t$ by $n$ is not the problem because this can be done with the help of the Bernoulli polynomials. What I ment was the differentiation of the Euler-MacLaurin sum formula for this sum (in your case: $O\left(n^{t-3}\right)$. $\endgroup$ – user90369 Sep 3 '16 at 19:39
  • 1
    $\begingroup$ Certainly, you can take the derivative of the polynomial whose values at the integers agrees with the sum, but taking the derivative of a sum with respect to the upper limit is not quite the same. The upper and lower limits of a sum generally differ by an integer. It is not immediately apparent how to define $\sum\limits_{k=1}^{5.2}k^2$. $\endgroup$ – robjohn Sep 6 '16 at 12:07
  • 1
    $\begingroup$ $\sum\limits_{k=1}^{5.2}k^2$ has no clear meaning by itself. We can find a polynomial that agrees with the sum when the upper limit is an integer, and apply that to $5.2$; however, there are many functions that agree with the sum when the upper limit is an integer. Why should the polynomial be chosen? $\endgroup$ – robjohn Sep 6 '16 at 12:35

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