View source for Bernoulli polynomials of the second kind

{{Use American English|date = March 2019}}
{{Short description|Polynomial sequence}}
The '''Bernoulli polynomials of the second kind'''<ref name="jordan2" /><ref name="jordan">{{cite book|last1=Jordan |first1=Charles |title=The Calculus of Finite Differences (3rd Edition) |year=1965 |publisher=Chelsea Publishing Company}}</ref> {{math|''ψ<sub>n</sub>''(''x'')}}, also known as the '''Fontana–Bessel polynomials''',<ref name="blagouchine">{{cite journal |last1= Blagouchine|first1= Iaroslav V.|title= Three notes on Ser's and Hasse's representations for the zeta-functions |journal= INTEGERS: The Electronic Journal of Combinatorial Number Theory |date= 2018 |volume= 18A|issue= #A3|pages= 1–45|url=http://math.colgate.edu/~integers/sjs3/sjs3.pdf}} [https://arxiv.org/abs/1606.02044 arXiv]</ref> are the polynomials defined by the following generating function:
<math display="block">
\frac{z(1+z)^x}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(x) ,\qquad |z|<1.
</math>

The first five polynomials are:
<math display="block">
\begin{align}
\psi_0(x) &=                                                           1  \\[2mm]
\psi_1(x) &=                                                       x + \frac{1}{2} \\[2mm]
\psi_2(x) &=                                      \frac{1}{2} x^2    - \frac{1}{12} \\[2mm]
\psi_3(x) &=                    \frac{1}{6} x^3 - \frac{1}{4} x^2    + \frac{1}{24} \\[2mm]
\psi_4(x) &= \frac{1}{24} x^4 - \frac{1}{6} x^3 + \frac{1}{6} x^2    - \frac{19}{720}
\end{align}
</math>

Some authors define these polynomials slightly differently<ref name="roman">{{cite book|last1=Roman |first1=S. |title=The Umbral Calculus |year=1984 |publisher=New York: Academic Press}}</ref><ref>{{cite book|last1=Weisstein |first1= Eric W. |title=Bernoulli Polynomial of the Second Kind |publisher=From MathWorld--A Wolfram Web Resource | url=http://mathworld.wolfram.com/BernoulliPolynomialoftheSecondKind.html}}</ref>
<math display="block"> \frac{z \left(1+z\right)^x}{\ln(1+z)} = \sum_{n=0}^\infty \frac{z^n}{n!} \psi^*_n(x) ,\qquad |z|<1, </math>
so that
<math> \psi^*_n(x) = \psi_n(x) \, n! </math>
and may also use a different notation for them (the most used alternative notation is {{math|''b<sub>n</sub>''(''x'')}}). Under this convention, the polynomials form a [[Sheffer sequence]].

The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan,<ref name="jordan2">{{cite journal |last1= Jordan|first1= Charles|title= Sur des polynomes analogues aux polynomes de Bernoulli, et sur des formules de sommation analogues à celle de Maclaurin-Euler |journal= Acta Sci. Math. (Szeged) |date= 1928 |volume= 4 |pages= 130–150}}</ref><ref name="jordan" /> but their history may also be traced back to the much earlier works.<ref name="blagouchine" />

==Integral representations==
The Bernoulli polynomials of the second kind may be represented via these integrals<ref name="jordan2" /><ref name="jordan" />
<math display="block">
\psi_{n}(x) = \int_x^{x+1}\! \binom{u}{n} \, du = \int_0^1 \binom{x+u}{n} \, du
</math>
as well as<ref name="blagouchine" />
<math display="block">\begin{align}
\psi_{n}(x) &= \frac{\left(-1\right)^{n+1}}{\pi} 
\int_0^\infty  \frac{\pi \cos\pi x - \sin\pi x \ln z}{(1+z)^n} \cdot\frac{z^x  dz}{\ln^2 z +\pi^2}
,\qquad  -1\leq x\leq n-1\, \\[3mm]
\psi_{n}(x) &= \frac{\left(-1\right)^{n+1}}{\pi} 
\int_{-\infty}^{+\infty} \frac{\pi \cos\pi x - v\sin\pi x }{\left(1+e^v\right)^n} \cdot \frac{e^{v(x+1)} }{v^2 +\pi^2}\, dv ,\qquad  -1\leq x\leq n-1\,
\end{align}</math>

These polynomials are, therefore, up to a constant, the [[antiderivative]] of the [[binomial coefficient]] and also that of the [[falling factorial]].<ref name="jordan2" /><ref name="jordan" /><ref name="blagouchine" />

==Explicit formula==
For an arbitrary {{math|''n''}}, these polynomials may be computed explicitly via the following summation formula<ref name="jordan2" /><ref name="jordan" /><ref name="blagouchine" />
<math display="block"> \psi_{n}(x) = \frac{1}{(n-1)!}\sum_{l=0}^{n-1} \frac{s(n-1,l)}{l+1} x^{l+1} + G_{n}, \qquad n=1,2,3,\ldots </math>
where {{math|''s''(''n'',''l'')}} are the signed [[Stirling numbers of the first kind]] and {{math|''G''<sub>''n''</sub>}} are the [[Gregory coefficients]].

The expansion of the Bernoulli polynomials of the second kind into a Newton series reads<ref name="jordan2" /><ref name="jordan" />
<math display="block">\psi_{n}(x) = G_0 \binom{x}{n} + G_1 \binom{x}{n-1} + G_2 \binom{x}{n-2} + \ldots + G_n</math>
It can be shown using the second integral representation and [[Vandermonde's identity]].

==Recurrence formula==

The Bernoulli polynomials of the second kind satisfy the recurrence relation<ref name="jordan2" /><ref name="jordan" />
<math display="block">\psi_{n}(x+1) - \psi_{n}(x) = \psi_{n-1}(x)</math>
or equivalently
<math display="block">\Delta\psi_{n}(x) = \psi_{n-1}(x)</math>

The repeated difference produces<ref name="jordan2" /><ref name="jordan" />
<math display="block">\Delta^m\psi_{n}(x) = \psi_{n-m}(x)</math>

==Symmetry property==
The main property of the symmetry reads<ref name="jordan" /><ref name="roman" />
<math display="block"> \psi_{n}{\left(\tfrac{1}{2}n-1+x\right)} = \left(-1\right)^n \psi_{n}{\left(\tfrac{1}{2}n-1-x\right)} </math>

==Some further properties and particular values==
Some properties and particular values of these polynomials include 
<math display="block">\begin{align}
&\psi_n(0) = G_n  \\[2mm]
&\psi_n(1) = G_{n-1} + G_{n} \\[2mm]
&\psi_n(-1) = \left(-1\right)^{n+1} \sum_{m=0}^n \left|G_m\right| = \left(-1\right)^n C_n\\[2mm]
&\psi_n(n-2) = - \left|G_n\right|  \\[2mm]
&\psi_n(n-1) = \left(-1\right)^n \psi_n(-1) = 1 - \sum_{m=1}^n \left|G_m\right| \\[2mm]
&\psi_{2n}(n-1) = M_{2n} \\[2mm]
&\psi_{2n}(n-1+y) = \psi_{2n}(n-1-y) \\[2mm]
&\psi_{2n+1}(n-\tfrac{1}{2}+y) = -\psi_{2n+1}(n-\tfrac{1}{2}-y) \\[2mm]
&\psi_{2n+1}(n-\tfrac{1}{2}) = 0 
\end{align}</math>
where {{math|''C''<sub>''n''</sub>}} are the ''Cauchy numbers of the second kind'' and {{math|''M''<sub>''n''</sub>}} are the ''central difference coefficients''.<ref name="jordan2" /><ref name="jordan" /><ref name="blagouchine" />

==Some series involving the Bernoulli polynomials of the second kind==
The [[digamma function]] {{math|Ψ(''x'')}} may be expanded into a series with the Bernoulli polynomials of the second kind
in the following way<ref name="blagouchine" />
<math display="block">
\Psi(v) = \ln(v+a) + \sum_{n=1}^\infty \frac{(-1)^n\psi_{n}(a)\,(n-1)!}{(v)_{n}},\qquad \Re(v) > -a,
</math>
and hence<ref name="blagouchine" />
<math display="block">\gamma= -\ln(a+1) - \sum_{n=1}^\infty\frac{(-1)^n \psi_{n}(a)}{n},\qquad \Re(a)>-1 </math>
and
<math display="block">\gamma = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n}\left\{\psi_{n}(a)+ \psi_{n}\left(-\frac{a}{1+a}\right)\right\},
\quad a>-1</math>
where {{math|''γ''}} is [[Euler's constant]]. Furthermore, we also have<ref name="blagouchine" />
<math display="block">
\Psi(v) = \frac{1}{v + a - \frac{1}{2}} \left\{\ln\Gamma(v+a) + v - \frac{1}{2}\ln(2\pi) - \frac{1}{2} + \sum_{n=1}^\infty \frac{\left(-1\right)^n \psi_{n+1}(a)}{(v)_{n}} \left(n-1\right)!\right\}, \quad \Re(v)>-a, 
</math>
where {{math|Γ(''x'')}} is the [[gamma function]]. The [[Hurwitz zeta function|Hurwitz]] and [[Riemann zeta function]]s may be expanded into these polynomials as follows<ref name="blagouchine" />
<math display="block">
\zeta(s,v) = \frac{(v+a)^{1-s} }{s-1} +  \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a)
\sum_{k=0}^{n} \left(-1\right)^k \binom{n}{k} (k+v)^{-s}
</math>
and
<math display="block">
\zeta(s)= \frac{(a+1)^{1-s} }{s-1} +  \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a)
\sum_{k=0}^{n} \left(-1\right)^k \binom{n}{k} (k+1)^{-s}
</math>
and also
<math display="block">
\zeta(s) = 1 + \frac{(a+2)^{1-s}}{s-1} +  \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a)
\sum_{k=0}^{n} \left(-1\right)^k \binom{n}{k} (k+2)^{-s}
</math>

The Bernoulli polynomials of the second kind are also involved in the following relationship<ref name="blagouchine" />
<math display="block">
\big(v+a-\tfrac{1}{2}\big)\zeta(s,v) = -\frac{\zeta(s-1,v+a)}{s-1} + \zeta(s-1,v) +  
 \sum_{n=0}^\infty \left(-1\right)^n \psi_{n+2}(a) \sum_{k=0}^{n} \left(-1\right)^k \binom{n}{k} (k+v)^{-s} 
</math>
between the zeta functions, as well as in various formulas for the [[Stieltjes constants]], e.g.<ref name="blagouchine" />
<math display="block">
\gamma_m(v) = -\frac{\ln^{m+1}(v+a)}{m+1} + \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a)
\sum_{k=0}^{n} \left(-1\right)^k \binom{n}{k}\frac{\ln^m (k+v)}{k+v}
</math>
and
<math display="block">
 \gamma_m(v)=\frac{1}{\tfrac{1}{2}-v-a}
\left\{\frac{(-1)^m}{m+1}\,\zeta^{(m+1)}(0,v+a)- (-1)^m \zeta^{(m)}(0,v)
- \sum_{n=0}^\infty (-1)^n \psi_{n+2}(a) 
\sum_{k=0}^{n} (-1)^k \binom{n}{k}\frac{\ln^m (k+v)}{k+v}\right\} 
</math>
which are both valid for <math>\Re(a) > -1</math> and <math>v \in \mathbb{C}\setminus\!\{0,-1,-2,\ldots\}</math>.

== See also==
{{div col}}
* [[Bernoulli polynomials]]
* [[Stirling polynomials]]
* [[Gregory coefficients]]
* [[Bernoulli numbers]]
* [[Difference polynomials]]
* [[Poly-Bernoulli number]]
* [[Mittag-Leffler polynomials]]
{{div col end}}

== References ==
{{Reflist}}

== Mathematics ==

[[Category:Polynomials]]
[[Category:Number theory]]