Английская Википедия:Bernoulli polynomials of the second kind
Шаблон:Use American English Шаблон:Short description The Bernoulli polynomials of the second kind[1][2] Шаблон:Math, also known as the Fontana-Bessel polynomials,[3] are the polynomials defined by the following generating function:
- <math>
\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>
\begin{array}{l} \displaystyle \psi_0(x)=1 \\[2mm] \displaystyle \psi_1(x)=x+\frac12 \\[2mm] \displaystyle \psi_2(x)=\frac12x^2-\frac{1}{12}\\[2mm] \displaystyle \psi_3(x)=\frac16x^3-\frac14x^2+\frac{1}{24}\\[2mm] \displaystyle \psi_4(x)=\frac{1}{24}x^4-\frac16x^3+\frac16x^2 -\frac{19}{720} \end{array} </math>
Some authors define these polynomials slightly differently[4][5]
- <math>
\frac{z(1+z)^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 Шаблон:Math). 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,[1][2] but their history may also be traced back to the much earlier works.[3]
Integral representations
The Bernoulli polynomials of the second kind may be represented via these integrals[1][2]
- <math>
\psi_{n}(x) = \int\limits_x^{x+1}\! \binom{u}{n}\, du = \int\limits_0^1 \binom{x+u}{n}\, du </math>
as well as[3]
- <math>
\begin{array}{l} \displaystyle \psi_{n}(x)=\frac{(-1)^{n+1}}{\pi} \int\limits_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] \displaystyle \psi_{n}(x)=\frac{(-1)^{n+1}}{\pi} \int\limits_{-\infty}^{+\infty} \frac{\pi \cos\pi x - v\sin\pi x }{\,(1+e^v)^n} \cdot\frac{e^{v(x+1)} }{v^2 +\pi^2}\, dv ,\qquad -1\leq x\leq n-1\, \end{array} </math>
These polynomials are, therefore, up to a constant, the antiderivative of the binomial coefficient and also that of the falling factorial.[1][2][3]
Explicit formula
For an arbitrary Шаблон:Math, these polynomials may be computed explicitly via the following summation formula[1][2][3]
- <math>
\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 Шаблон:Math are the signed Stirling numbers of the first kind and Шаблон:Math are the Gregory coefficients.
The expansion of the Bernoulli polynomials of the second kind into a Newton series reads[1][2]
- <math>
\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[1][2]
- <math>
\psi_{n}(x+1) - \psi_{n}(x) = \psi_{n-1}(x) </math>
or equivalently
- <math>
\Delta\psi_{n}(x) = \psi_{n-1}(x) </math>
The repeated difference produces[1][2]
- <math>
\Delta^m\psi_{n}(x) = \psi_{n-m}(x) </math>
Symmetry property
The main property of the symmetry reads[2][4]
- <math>
\psi_{n}(\tfrac12n-1+x) = (-1)^n\psi_{n}(\tfrac12n-1-x) </math>
Some further properties and particular values
Some properties and particular values of these polynomials include
- <math>
\begin{array}{l} \displaystyle \psi_n(0)=G_n \\[2mm] \displaystyle \psi_n(1)=G_{n-1} + G_{n} \\[2mm] \displaystyle \psi_n(-1)= (-1)^{n+1} \sum_{m=0}^n |G_m| = (-1)^n C_n\\[2mm] \displaystyle \psi_n(n-2)=-|G_n| \\[2mm] \displaystyle \psi_n(n-1)= (-1)^n \psi_n(-1) = 1- \sum_{m=1}^n |G_m|\\[2mm] \displaystyle \psi_{2n}(n-1)=M_{2n} \\[2mm] \displaystyle \psi_{2n}(n-1+y)=\psi_{2n}(n-1-y) \\[2mm] \displaystyle \psi_{2n+1}(n-\tfrac12+y)=-\psi_{2n+1}(n-\tfrac12-y) \\[2mm] \displaystyle \psi_{2n+1}(n-\tfrac12)=0 \end{array} </math>
where Шаблон:Math are the Cauchy numbers of the second kind and Шаблон:Math are the central difference coefficients.[1][2][3]
Some series involving the Bernoulli polynomials of the second kind
The digamma function Шаблон:Math may be expanded into a series with the Bernoulli polynomials of the second kind in the following way[3]
- <math>
\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[3]
<math>\gamma= -\ln(a+1) - \sum_{n=1}^\infty\frac{(-1)^n \psi_{n}(a)}{n},\qquad \Re(a)>-1 </math>
and
- <math>\gamma=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n}\Big\{\psi_{n}(a)+ \psi_{n}\Big(-\frac{a}{1+a}\Big)\Big\},
\quad a>-1</math>
where Шаблон:Math is Euler's constant. Furthermore, we also have[3]
- <math>
\Psi(v)= \frac{1}{v+a-\tfrac12}\left\{\ln\Gamma(v+a) + v - \frac12\ln2\pi - \frac12 + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{(v)_{n}}(n-1)!\right\},\qquad \Re(v)>-a, </math>
where Шаблон:Math is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows[3]
- <math>
\zeta(s,v)= \frac{(v+a)^{1-s} }{s-1} + \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a) \sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+v)^{-s} </math>
and
- <math>
\zeta(s)= \frac{(a+1)^{1-s} }{s-1} + \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a) \sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+1)^{-s} </math>
and also
- <math>
\zeta(s) =1 + \frac{(a+2)^{1-s}}{s-1} + \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a) \sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+2)^{-s} </math>
The Bernoulli polynomials of the second kind are also involved in the following relationship[3]
- <math>
\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 (-1)^n \psi_{n+2}(a) \sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+v)^{-s}
</math>
between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.[3]
- <math>
\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} (-1)^k \binom{n}{k}\frac{\ln^m (k+v)}{k+v} </math>
and
- <math>
\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
- Bernoulli polynomials
- Stirling polynomials
- Gregory coefficients
- Bernoulli numbers
- Difference polynomials
- Poly-Bernoulli number
- Mittag-Leffler polynomials
References
Mathematics
