Bernoulli polynomials of the second kind

The Bernoulli polynomials of the second kind $ψ_{n}(x)$, also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating function: $$ \frac{z(1+z)^x}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(x) ,\qquad |z|<1. $$

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

Some authors define these polynomials slightly differently $$ \frac{z \left(1+z\right)^x}{\ln(1+z)} = \sum_{n=0}^\infty \frac{z^n}{n!} \psi^*_n(x) ,\qquad |z|<1, $$ so that $$ \psi^*_n(x) = \psi_n(x) \, n! $$ and may also use a different notation for them (the most used alternative notation is $b_{n}(x)$). 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, but their history may also be traced back to the much earlier works.

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

These polynomials are, therefore, up to a constant, the antiderivative of the binomial coefficient and also that of the falling factorial.

Explicit formula
For an arbitrary $n$, these polynomials may be computed explicitly via the following summation formula $$ \psi_{n}(x) = \frac{1}{(n-1)!}\sum_{l=0}^{n-1} \frac{s(n-1,l)}{l+1} x^{l+1} + G_{n}, \qquad n=1,2,3,\ldots $$ where $s(n,l)$ are the signed Stirling numbers of the first kind and $G_{n}$ are the Gregory coefficients.

The expansion of the Bernoulli polynomials of the second kind into a Newton series reads $$\psi_{n}(x) = G_0 \binom{x}{n} + G_1 \binom{x}{n-1} + G_2 \binom{x}{n-2} + \ldots + G_n$$ 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 $$\psi_{n}(x+1) - \psi_{n}(x) = \psi_{n-1}(x)$$ or equivalently $$\Delta\psi_{n}(x) = \psi_{n-1}(x)$$

The repeated difference produces $$\Delta^m\psi_{n}(x) = \psi_{n-m}(x)$$

Symmetry property
The main property of the symmetry reads $$ \psi_{n}{\left(\tfrac{1}{2}n-1+x\right)} = \left(-1\right)^n \psi_{n}{\left(\tfrac{1}{2}n-1-x\right)} $$

Some further properties and particular values
Some properties and particular values of these polynomials include $$\begin{align} &\psi_n(0) = G_n \\[2mm] &\psi_n(1) = G_{n-1} + G_{n} \\[2mm] &\psi_n(-1) = \left(-1\right)^{n+1} \sum_{m=0}^n \left|G_m\right| = \left(-1\right)^n C_n\\[2mm] &\psi_n(n-2) = - \left|G_n\right| \\[2mm] &\psi_n(n-1) = \left(-1\right)^n \psi_n(-1) = 1 - \sum_{m=1}^n \left|G_m\right| \\[2mm] &\psi_{2n}(n-1) = M_{2n} \\[2mm] &\psi_{2n}(n-1+y) = \psi_{2n}(n-1-y) \\[2mm] &\psi_{2n+1}(n-\tfrac{1}{2}+y) = -\psi_{2n+1}(n-\tfrac{1}{2}-y) \\[2mm] &\psi_{2n+1}(n-\tfrac{1}{2}) = 0 \end{align}$$ where $C_{n}$ are the Cauchy numbers of the second kind and $M_{n}$ are the central difference coefficients.

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

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