Bernoulli polynomials of the second kind information

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

The first five polynomials are: ${\begin{aligned}\psi _{0}(x)&=1\\[2mm]\psi _{1}(x)&=x+{\frac {1}{2}}\\[2mm]\psi _{2}(x)&={\frac {1}{2}}x^{2}-{\frac {1}{12}}\\[2mm]\psi _{3}(x)&={\frac {1}{6}}x^{3}-{\frac {1}{4}}x^{2}+{\frac {1}{24}}\\[2mm]\psi _{4}(x)&={\frac {1}{24}}x^{4}-{\frac {1}{6}}x^{3}+{\frac {1}{6}}x^{2}-{\frac {19}{720}}\end{aligned}}$

Some authors define these polynomials slightly differently^[4]^[5] ${\frac {z\left(1+z\right)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\psi _{n}^{*}(x),\qquad |z|<1,$ 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,^[1]^[2] but their history may also be traced back to the much earlier works.^[3]

^ ^a ^b Jordan, Charles (1928). "Sur des polynomes analogues aux polynomes de Bernoulli, et sur des formules de sommation analogues à celle de Maclaurin-Euler". Acta Sci. Math. (Szeged). 4: 130–150.
^ ^a ^b Jordan, Charles (1965). The Calculus of Finite Differences (3rd Edition). Chelsea Publishing Company.
^ ^a ^b Blagouchine, Iaroslav V. (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A (#A3): 1–45. arXiv
^ Roman, S. (1984). The Umbral Calculus. New York: Academic Press.
^ Weisstein, Eric W. Bernoulli Polynomial of the Second Kind. From MathWorld--A Wolfram Web Resource.

[jordan2-1] Jordan, Charles (1928). "Sur des polynomes analogues aux polynomes de Bernoulli, et sur des formules de sommation analogues à celle de Maclaurin-Euler". Acta Sci. Math. (Szeged). 4: 130–150.

[jordan-2] Jordan, Charles (1965). The Calculus of Finite Differences (3rd Edition). Chelsea Publishing Company.

[blagouchine-3] Blagouchine, Iaroslav V. (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A (#A3): 1–45. arXiv

[roman-4] Roman, S. (1984). The Umbral Calculus. New York: Academic Press.

[5] Weisstein, Eric W. Bernoulli Polynomial of the Second Kind. From MathWorld--A Wolfram Web Resource.

Bernoulli polynomials of the second kind information

and 23 Related for: Bernoulli polynomials of the second kind information

Bernoulli polynomials of the second kind

Bernoulli polynomials

Digamma function

Bernoulli number

Stirling polynomials

Gregory coefficients

Sheffer sequence

Bessel function

Stirling numbers of the second kind

Difference polynomials

Bernstein polynomial

Stirling numbers of the first kind

List of things named after Leonhard Euler

Eulerian number

Stieltjes constants

Euler numbers

Taylor series

Fundamental theorem of algebra

Differential equation

Symmetry of second derivatives

List of triangle topics

Figurate number

Gamma function