define , equivalently . Then has the following closed form in umbral notation, replacing by : , e.g. Note that by virtue of the logarithmic derivative of Riemann's functional equation, taken after applying Euler's reflection formula, these expressions in terms of the can be written in terms of , e.g. 6. For , the same integral diverges because the integrand behaves like for . But the difference of two such integrals with corresponding degree differences is well-defined and exhibits very similar patterns, e.g. . See also Bernoulli polynomials of the second kind Stirling polynomials Poly-Bernoulli number References Polynomials