Bernoulli polynomials of the second kind

The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating function: z ( 1 + z ) x ln ⁡ ( 1 + z ) = ∑ n = 0 ∞ z n ψ n ( x ) , | z | < 1. {\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: ψ 0 ( x ) = 1 ψ 1 ( x ) = x + 1 2 ψ 2 ( x ) = 1 2 x 2 − 1 12 ψ 3 ( x ) = 1 6 x 3 − 1 4 x 2 + 1 24 ψ 4 ( x ) = 1 24 x 4 − 1 6 x 3 + 1 6 x 2 − 19 720 {\displaystyle {\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 z ( 1 + z ) x ln ⁡ ( 1 + z ) = ∑ n = 0 ∞ z n n !

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