is polynomially bounded on the right hand side of the complex plane, then the contour may be extended to infinity on the right hand side, allowing the transform to be written as where the constant c is to the left of α.
The Poisson–Mellin–Newton cycle, noted by Flajolet et al. in 1985, is the observation that the resemblance of the Nørlund–Rice integral to the Mellin transform is not accidental, but is related by means of the binomial transform and the Newton series.
be a sequence, and let g(t) be the corresponding Poisson generating function, that is, let Taking its Mellin transform one can then regain the original sequence by means of the Nörlund–Rice integral (see References "Mellin, seen from the sky"): where Γ is the gamma function which cancels with the gamma from Ramanujan's Master Theorem.
A closely related integral frequently occurs in the discussion of Riesz means.
Very roughly, it can be said to be related to the Nörlund–Rice integral in the same way that Perron's formula is related to the Mellin transform: rather than dealing with infinite series, it deals with finite series.
The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.