Krull-Webster’s theory offered an elegant extension of Bohr-Mollerup’s theorem and has proved to be a very nice and useful contribution to the resolution of the difference equation Δf = g on the real half-line \(\mathbb {R}_+\). In this book, we have provided a significant generalization of Krull-Webster’s theory by considerably relaxing the asymptotic condition imposed on the function g, and we have demonstrated through various examples how this generalization provides a unified framework to investigate the properties of many functions. This framework has indeed enabled us to derive several general formulas that now constitute a powerful toolbox and even a genuine Swiss Army knife to investigate a large variety of functions.

The key point of this generalization was the discovery of expression (1.4) for the sequence \(n\mapsto f^p_n[g](x)\) for any \(p\in \mathbb {N}\). We also observe that our uniqueness and existence results strongly rely on Lemma 2.7 together with identities (3.3) and (3.8). These results actually constitute the common core and even the fundamental cornerstone of all the subsequent formulas that we derived in this book. For instance, the generalized Stirling formula (6.21) has been obtained almost miraculously by merely integrating both sides of the inequality given in Lemma 2.7 (see Remark 6.16). Similarly, Gregory’s summation formula (6.33) has been derived instantly by integrating both sides of identity (3.8), and we have shown how its remainder can be controlled using Lemma 2.7 again.

Our results clearly shed light on the way many of the classical special functions, such as the polygamma functions and the higher order derivatives of the Hurwitz zeta function, can be systematically studied, sometimes by deriving identities and formulas almost mechanically.

Beyond this systematization aspect, our theory has enabled us to introduce a number of new important and useful objects. For instance, the map Σ itself is a new concept that appears to be as fundamental as the basic antiderivative operation (cf. Definition 5.4). Both concepts are actually strongly related through, e.g., Propositions 6.19, 6.20, and 8.18. Other concepts such as the asymptotic constant and the generalized Binet function also appear to be new fundamental objects that merit further study. For instance, it is remarkable that the asymptotic constant appears not only in the generalized Stirling formula (Theorem 6.13), but also in many other important formulas, such as the generalized Euler constant (Proposition 6.36), the Weierstrassian form (Theorem 8.7), the analogue of Raabe’s formula (Proposition 8.18), the analogue of Gauss’ multiplication formula (Proposition 8.28), the asymptotic expansion (Proposition 8.36), and the generalized Liu formula (Proposition 8.42).

Our work has also revealed how important and natural are the higher order convexity properties. Although these properties seem to be still poorly used in mathematical analysis, they actually constitute an essential and highly useful ingredient in the development of our theory and therefore also merit further investigation (see, e.g., Proposition 4.14 and Remark 4.15).

In conclusion, the results that we have obtained as well as the new concepts that we have introduced and explored show that this area of investigation is very rich and intriguing. We have just skimmed the surface, and there are a lot of questions that emerge naturally. We now list some below.

  • Find a simple characterization of the domain of the map Σ (see Proposition 5.21 and Conjecture 5.23).

  • Find necessary and sufficient conditions on the function g to ensure both the uniqueness and existence of solutions lying in \(\mathcal {K}^p\) to the equation Δf = g (cf. Webster’s question in Appendix C).

  • Find a natural extension of the map Σ to a larger domain, e.g., a real linear space of functions that would include not only the current admissible functions but also every function that has an exponential growth.

  • Find a general method to determine a simple or compact expression for the asymptotic constant σ[g] associated with any function g lying in \(\mathcal {C}^0\cap \mathrm {dom}(\Sigma )\) (cf. our discussion in Sect. 8.5).

  • Find general methods to determine analogues of Euler’s reflection formula (cf. our discussion on Herglotz’s trick in Sect. 8.9) and Gauss’ digamma theorem for any multiple \(\log \Gamma \)-type function.

  • Find necessary and sufficient conditions on the function g for the function Σg to be of class \(\mathcal {C}^{\infty }\) or even real analytic.

  • Find an extension of our theory to functions of a complex variable. On this issue, it is noteworthy that a very nice “complex” counterpart of Bohr-Mollerup’s characterization of the gamma function was established by Wielandt (see, e.g., Srinivasan [92] and Srivastava and Choi [93, p. 12] and the references therein).

  • Find an extension of our theory by replacing the equation Δf = g with the more general first-order linear difference equation

    $$\displaystyle \begin{aligned} f(x+1)-a{\,}f(x) ~=~ g(x), \end{aligned}$$

    where a is a given constant. Consider also linear difference equations of any order. Partial results along this line can be found, e.g., in John [49, Theorem C].