The Bernoulli polynomials of the second kind^[1][2] ψ_n(x), also known as the Fontana-Bessel polynomials,^[3] are the polynomials defined by the following generating function:

${\displaystyle {\frac {z(1+z)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(x),\qquad |z|<1.}$

The first five polynomials are:

${\displaystyle {\begin{array}{l}\displaystyle \psi _{0}(x)=1\\[2mm]\displaystyle \psi _{1}(x)=x+{\frac {1}{2}}\\[2mm]\displaystyle \psi _{2}(x)={\frac {1}{2}}x^{2}-{\frac {1}{12}}\\[2mm]\displaystyle \psi _{3}(x)={\frac {1}{6}}x^{3}-{\frac {1}{4}}x^{2}+{\frac {1}{24}}\\[2mm]\displaystyle \psi _{4}(x)={\frac {1}{24}}x^{4}-{\frac {1}{6}}x^{3}+{\frac {1}{6}}x^{2}-{\frac {19}{720}}\end{array}}}$

Some authors define these polynomials slightly differently^[4][5]

${\displaystyle {\frac {z(1+z)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\psi _{n}^{*}(x),\qquad |z|<1,}$

so that

${\displaystyle \psi _{n}^{*}(x)=\psi _{n}(x)\,n!}$

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,^[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]

${\displaystyle \psi _{n}(x)=\int \limits _{x}^{x+1}\!{\binom {u}{n}}\,du=\int \limits _{0}^{1}{\binom {x+u}{n}}\,du}$

as well as^[3]

${\displaystyle {\begin{array}{l}\displaystyle \psi _{n}(x)={\frac {(-1)^{n+1}}{\pi }}\int \limits _{0}^{\infty }{\frac {\pi \cos \pi x-\sin \pi x\ln z}{(1+z)^{n}}}\cdot {\frac {z^{x}dz}{\ln ^{2}z+\pi ^{2}}},\qquad -1\leq x\leq n-1\,\\[3mm]\displaystyle \psi _{n}(x)={\frac {(-1)^{n+1}}{\pi }}\int \limits _{-\infty }^{+\infty }{\frac {\pi \cos \pi x-v\sin \pi x}{\,(1+e^{v})^{n}}}\cdot {\frac {e^{v(x+1)}}{v^{2}+\pi ^{2}}}\,dv,\qquad -1\leq x\leq n-1\,\end{array}}}$

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 n, these polynomials may be computed explicitly via the following summation formula^[1][2][3]

${\displaystyle \psi _{n}(x)={\frac {1}{(n-1)!}}\sum _{l=0}^{n-1}{\frac {s(n-1,l)}{l+1}}x^{l+1}+G_{n},\qquad n=1,2,3,\ldots }$

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^[1][2]

${\displaystyle \psi _{n}(x)=G_{0}{\binom {x}{n}}+G_{1}{\binom {x}{n-1}}+G_{2}{\binom {x}{n-2}}+\ldots +G_{n}}$

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^[1][2]

${\displaystyle \psi _{n}(x+1)-\psi _{n}(x)=\psi _{n-1}(x)}$

or equivalently

${\displaystyle \Delta \psi _{n}(x)=\psi _{n-1}(x)}$

The repeated difference produces^[1][2]

${\displaystyle \Delta ^{m}\psi _{n}(x)=\psi _{n-m}(x)}$

Symmetry property

The main property of the symmetry reads^[2][4]

${\displaystyle \psi _{n}({\tfrac {1}{2}}n-1+x)=(-1)^{n}\psi _{n}({\tfrac {1}{2}}n-1-x)}$

Some further properties and particular values

Some properties and particular values of these polynomials include

${\displaystyle {\begin{array}{l}\displaystyle \psi _{n}(0)=G_{n}\\[2mm]\displaystyle \psi _{n}(1)=G_{n-1}+G_{n}\\[2mm]\displaystyle \psi _{n}(-1)=(-1)^{n+1}\sum _{m=0}^{n}|G_{m}|=(-1)^{n}C_{n}\\[2mm]\displaystyle \psi _{n}(n-2)=-|G_{n}|\\[2mm]\displaystyle \psi _{n}(n-1)=(-1)^{n}\psi _{n}(-1)=1-\sum _{m=1}^{n}|G_{m}|\\[2mm]\displaystyle \psi _{2n}(n-1)=M_{2n}\\[2mm]\displaystyle \psi _{2n}(n-1+y)=\psi _{2n}(n-1-y)\\[2mm]\displaystyle \psi _{2n+1}(n-{\tfrac {1}{2}}+y)=-\psi _{2n+1}(n-{\tfrac {1}{2}}-y)\\[2mm]\displaystyle \psi _{2n+1}(n-{\tfrac {1}{2}})=0\end{array}}}$

where C_n are the Cauchy numbers of the second kind and M_n are the central difference coefficients.^[1][2][3]

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^[3]

${\displaystyle \Psi (v)=\ln(v+a)+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)\,(n-1)!}{(v)_{n}}},\qquad \Re (v)>-a,}$

and hence^[3]

${\displaystyle \gamma =-\ln(a+1)-\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)}{n}},\qquad \Re (a)>-1}$

and

${\displaystyle \gamma =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n}}{\Big \{}\psi _{n}(a)+\psi _{n}{\Big (}-{\frac {a}{1+a}}{\Big )}{\Big \}},\quad a>-1}$

where γ is Euler's constant. Furthermore, we also have^[3]

${\displaystyle \Psi (v)={\frac {1}{v+a-{\tfrac {1}{2}}}}\left\{\ln \Gamma (v+a)+v-{\frac {1}{2}}\ln 2\pi -{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n+1}(a)}{(v)_{n}}}(n-1)!\right\},\qquad \Re (v)>-a,}$

where Γ(x) is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows^[3]

${\displaystyle \zeta (s,v)={\frac {(v+a)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+v)^{-s}}$

and

${\displaystyle \zeta (s)={\frac {(a+1)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s}}$

and also

${\displaystyle \zeta (s)=1+{\frac {(a+2)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+2)^{-s}}$

The Bernoulli polynomials of the second kind are also involved in the following relationship^[3]

${\displaystyle {\big (}v+a-{\tfrac {1}{2}}{\big )}\zeta (s,v)=-{\frac {\zeta (s-1,v+a)}{s-1}}+\zeta (s-1,v)+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+v)^{-s}}$

between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.^[3]

${\displaystyle \gamma _{m}(v)=-{\frac {\ln ^{m+1}(v+a)}{m+1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}}$

and

${\displaystyle \gamma _{m}(v)={\frac {1}{{\tfrac {1}{2}}-v-a}}\left\{{\frac {(-1)^{m}}{m+1}}\,\zeta ^{(m+1)}(0,v+a)-(-1)^{m}\zeta ^{(m)}(0,v)-\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}\right\}}$

which are both valid for ${\displaystyle \Re (a)>-1}$ and ${\displaystyle v\in \mathbb {C} \setminus \!\{0,-1,-2,\ldots \}}$.

See also

• Bernoulli polynomials
• Stirling polynomials
• Gregory coefficients
• Bernoulli numbers
• Difference polynomials
• Poly-Bernoulli number
• Mittag-Leffler polynomials

References

Mathematics