Uniqueness and Existence Results

Abstract

In this chapter, we establish Theorems 1.4 and 1.5 and show that, under the assumptions of these theorems, the sequence $n\mapsto f^p_n[g]$ converges uniformly on any bounded subset of $\mathbb {R}_+$. We also discuss the particular case where the sequence n↦g(n) is summable. Lastly, we provide historical notes on Krull-Webster’s theory and some of its improvements.

In this chapter, we establish Theorems 1.4 and 1.5 and show that, under the assumptions of these theorems, the sequence $n\mapsto f^p_n[g]$ converges uniformly on any bounded subset of $\mathbb {R}_+$. We also discuss the particular case where the sequence n↦g(n) is summable. Lastly, we provide historical notes on Krull-Webster’s theory and some of its improvements.

Although their proofs are short and elementary, the main results given in this chapter are of utmost importance. They constitute the fundamental cornerstone of the whole theory developed in this book.

3.1 Main Results

We start this chapter by establishing a slightly improved version of our uniqueness Theorem 1.5. We state this new version in Theorem 3.1 below and provide a very short proof. Let us first note that any solution $f\colon \mathbb {R}_+\to \mathbb {R}$ to the equation Δf = g satisfies trivially the equations

$$\displaystyle \begin{aligned} \begin{array}{rcl} f(n) & =&\displaystyle f(1) +\sum_{k=1}^{n-1}g(k),\qquad n\in\mathbb{N}^*;{} \end{array} \end{aligned} $$

(3.1)

$$\displaystyle \begin{aligned} \begin{array}{rcl} f(x+n) & =&\displaystyle f(x) + \sum_{k=0}^{n-1}g(x+k),\qquad n\in\mathbb{N}.{} \end{array} \end{aligned} $$

(3.2)

Moreover, using (1.4), (1.7), (3.1), and (3.2), we can easily derive the identity

$$\displaystyle \begin{aligned} f(x) ~=~ f(1) + f^p_n[g](x) + \rho^{p+1}_n[f](x),\qquad n\in\mathbb{N}^*. \end{aligned} $$

(3.3)

We also observe that the identity obtained by setting p = 0 in (3.3) can also be derived by subtracting (3.2) from (3.1).

Theorem 3.1 (Uniqueness)

Let $p\in \mathbb {N}$ and $g\in \mathcal {D}^p_{\mathrm {S}}$ . Suppose that $f\colon \mathbb {R}_+\to \mathbb {R}$ is a solution to the equation Δf = g that lies in $\mathcal {K}^p$ . Then, the following assertions hold.

• (a) We have that $f\in \mathcal {R}^{p+1}_{\mathrm {S}}$.

• (b) For each x > 0, the sequence $n\mapsto f^p_n[g](x)$ converges and we have

  $$\displaystyle \begin{aligned} f(x) ~=~ f(1) + \lim_{n\to\infty}f^p_n[g](x){\,}. \end{aligned}$$

• (c) The sequence $n\mapsto f^p_n[g]$ converges uniformly on any bounded subset of $\mathbb {R}_+$ to f − f(1).

Proof

We clearly have that $f\in \mathcal {D}^{p+1}_{\mathrm {S}}$. Assertion (a) then follows from Lemma 2.7 and the squeeze theorem. Assertion (b) follows from assertion (a) and identity (3.3). Now, let E be any bounded subset of $\mathbb {R}_+$. Using again identity (3.3) and Lemma 2.7, for large integer n we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sup_{x\in E}\left|f^p_n[g](x)-f(x)+f(1)\right| & =&\displaystyle \sup_{x\in E}\left|\rho^{p+1}_n[f](x)\right|\\ & \leq &\displaystyle \sup_{x\in E}\left|{\textstyle{{{x-1}\choose{p}}}}\right|~\sum_{j=0}^{\lceil \sup E\rceil -1}\left|\Delta^{p+1}f(n+j)\right|. \end{array} \end{aligned} $$

This establishes assertion (c). □

Example 3.2

Using Theorem 3.1 with $g(x)=\ln x$ and p = 1, we obtain that all solutions $f\colon \mathbb {R}_+\to \mathbb {R}$ lying in $\mathcal {K}^1$ to the equation $\Delta f(x)=\ln x$ are of the form $f(x)=c+\ln \Gamma (x)$, where $c\in \mathbb {R}$. We thus simply retrieve both Bohr-Mollerup’s Theorem 1.1 and Gauss’ limit (1.6), as expected. We also observe that the set $\mathcal {K}^1$ cannot be replaced with $\mathcal {K}^0$ in this characterization. For example, the function

$$\displaystyle \begin{aligned} f(x) ~=~ \textstyle{\ln\Gamma (x)+\ln(1+\frac{1}{2}\sin{}(2\pi x))} \end{aligned}$$

is also a solution lying in $\mathcal {K}^0$ to the equation $\Delta f(x)=\ln x$. $\lozenge$

Remark 3.3

We note that the assumption that $\ln f$ is convex in Bohr-Mollerup’s Theorem 1.1 can be easily replaced with the fact that $\ln f$ lies in $\mathcal {K}^1_+$ (without using the uniqueness Theorem 3.1). Indeed, if $\ln f$ is convex on [n, ∞) for some $n\in \mathbb {N}$, then using (3.2) we have that

$$\displaystyle \begin{aligned} \ln f(x) ~=~ \ln f(x+n)-\sum_{k=0}^{n-1}\ln(x+k),\qquad x>0, \end{aligned}$$

and hence $\ln f$ must be convex on $\mathbb {R}_+$ (as a finite sum of convex functions on $\mathbb {R}_+$). We can also replace $\mathcal {K}^1_+$ with $\mathcal {K}^1$; indeed, assuming that $\ln f$ lies in $\mathcal {K}^1_-$, we would obtain that $\Delta \ln f(x)=\ln x$ lies in $\mathcal {K}^0_-$ by Lemma 2.6(b), a contradiction. $\lozenge$

Remark 3.4 (A Proof of Bohr-Mollerup’s Theorem)

We have seen in Example 3.2 how both Bohr-Mollerup’s theorem and Gauss’ limit can be retrieved using our results. Let us now examine our proof in a self-contained way, using the needed arguments only. Let $f\colon \mathbb {R}_+\to \mathbb {R}$ be an eventually convex solution to the equation $\Delta f(x)=\ln x$. The nature of this equation shows that it is actually enough to assume that x > 1 to find the form of f(x). For any $n\in \mathbb {N}^*$ and any x > 1, we then have

$$\displaystyle \begin{aligned} f(n) ~=~ f(1)+\sum_{k=1}^{n-1}\ln k\qquad \mbox{and}\qquad f(x+n) ~=~ f(x)+\sum_{k=0}^{n-1}\ln(x+k) \end{aligned}$$

and hence also the identity

$$\displaystyle \begin{aligned} f(x) ~=~ f(1)+\left(\sum_{k=1}^{n-1}\ln k-\sum_{k=0}^{n-1}\ln(x+k)+x\ln n\right)+\rho_n(x), \end{aligned}$$

where

$$\displaystyle \begin{aligned} \rho_n(x) ~=~ f(x+n)-f(n)-x\ln n. \end{aligned}$$

To conclude the proof, we only need to show that, for each x > 1, the sequence n↦ρ _n(x) converges to zero. Let $n\in \mathbb {N}^*$ be so that f is convex on [n, ∞). Using the convexity of f we then obtain the following two inequalities

$$\displaystyle \begin{aligned} \begin{array}{rcl} f(n+1) & \leq &\displaystyle \textstyle{(1-\frac{1}{x}){\,}f(n)+\frac{1}{x}{\,}f(x+n)}{\,},\\ f(n+x) & \leq &\displaystyle \textstyle{\frac{1}{x}{\,}f(n+1)+(1-\frac{1}{x}){\,}f(x+n+1)}{\,}. \end{array} \end{aligned} $$

Using these inequalities and the identity $f(n+1)-f(n)=\ln n$, we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} 0 & \leq &\displaystyle \rho_n(x) ~=~ f(x+n)-f(n+1)-(x-1)\ln n\\ & \leq &\displaystyle (x-1){\,}(f(x+n+1)-f(x+n)-\ln n) ~=~ (x-1){\,}\textstyle{\ln(1+\frac{x}{n})}. \end{array} \end{aligned} $$

The proof is now complete since the latter expression converges to zero as n →∞. This shows to which extent the proofs of Bohr-Mollerup’s theorem and Gauss’ limit can be short and elementary. Note that a variant of this proof can be derived from the proof of Webster’s uniqueness theorem [98, Theorem 3.1]. $\lozenge$

Now that we have established the uniqueness Theorem 3.1, let us prepare the ground for the existence theorem. Using the definition of $\rho ^p_a[g](x)$ given in (1.7), we can easily derive the following two identities

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_a^p[g](p) & =&\displaystyle \Delta^pg(a){\,};{} \end{array} \end{aligned} $$

(3.4)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho^p_a[g](x)-\rho^{p+1}_a[g](x) & =&\displaystyle {\textstyle{{{x}\choose{p}}}}\,\rho^p_a[g](p){\,}.{} \end{array} \end{aligned} $$

(3.5)

These identities clearly show that the inclusions $\mathcal {R}^p_{\mathrm {S}}\subset \mathcal {D}^p_{\mathrm {S}}$ and $\mathcal {R}^p_{\mathrm {S}}\subset \mathcal {R}^{p+1}_{\mathrm {S}}$ hold for any $p\in \mathbb {N}$. We will see in Proposition 4.2 that these inclusions are actually strict.

Now, the following straightforward identities will also be useful as we continue

$$\displaystyle \begin{aligned} \begin{array}{rcl} f^p_{n+1}[g](x)-f^p_n[g](x) & =&\displaystyle -\rho^{p+1}_n[g](x){\,};{} \end{array} \end{aligned} $$

(3.6)

$$\displaystyle \begin{aligned} \begin{array}{rcl} f^p_n[g](x+1)-f^p_n[g](x) & =&\displaystyle g(x) - \rho^p_n[g](x){\,}.{} \end{array} \end{aligned} $$

(3.7)

For any integers 1 ≤ m ≤ n, from (3.6) we obtain

$$\displaystyle \begin{aligned} f^p_n[g](x) ~=~ f^p_m[g](x) -\sum_{k=m}^{n-1}\rho_k^{p+1}[g](x){\,}, \end{aligned} $$

(3.8)

which shows that, for any x > 0, the convergence of the sequence $n\mapsto f^p_n[g](x)$ is equivalent to the summability of the sequence $n\mapsto \rho ^{p+1}_n[g](x)$.

We now establish a slightly improved version of our existence Theorem 1.4. We first present a technical lemma, which follows straightforwardly from Lemma 2.7.

Lemma 3.5

Let $p\in \mathbb {N}$, $g\in \mathcal {K}^p$ , and $m\in \mathbb {N}^*$ be so that g is p-convex or p-concave on [m, ∞). Then, for any x ≥ 0 and any integer n ≥ m, we have

$$\displaystyle \begin{aligned} \left|\sum_{k=m}^{n-1}\rho^{p+1}_k[g](x)\right| ~\leq ~ \left|{\textstyle{{{x-1}\choose{p}}}}\right|\sum_{j=0}^{\lceil x\rceil -1}|\Delta^pg(n+j)-\Delta^pg(m+j)|. \end{aligned}$$

Proof

For any fixed x ≥ 0, the sequence $k\mapsto \rho ^{p+1}_k[g](x)$ for k ≥ m does not change in sign by Lemma 2.7 and hence we have

$$\displaystyle \begin{aligned} \left|\sum_{k=m}^{n-1}\rho^{p+1}_k[g](x)\right| ~=~ \sum_{k=m}^{n-1}\left|\rho^{p+1}_k[g](x)\right| ~\leq ~ \left|{\textstyle{{{x-1}\choose{p}}}}\right|\sum_{j=0}^{\lceil x\rceil -1}\left|\sum_{k=m}^{n-1}\Delta^{p+1}g(k+j)\right|, \end{aligned}$$

where the inner sum clearly telescopes to Δ^p g(n + j) − Δ^p g(m + j). □

Theorem 3.6 (Existence)

Let $p\in \mathbb {N}$ and $g\in \mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$ . The following assertions hold.

• (a) We have that $g\in \mathcal {R}^p_{\mathrm {S}}$ .

• (b) The sequence $n\mapsto f^p_n[g](x)$ converges for every x > 0, and the function $f\colon \mathbb {R}_+\to \mathbb {R}$ defined by

  $$\displaystyle \begin{aligned} f(x) ~=~ \lim_{n\to\infty} f^p_n[g](x){\,},\qquad x>0, \end{aligned}$$

  is a solution to the equation Δf = g that is p-concave (resp. p-convex) on any unbounded subinterval I of $\mathbb {R}_+$ on which g is p-convex (resp. p-concave). Moreover, we have f(1) = 0 and

  $$\displaystyle \begin{aligned} |f^p_n[g](x)-f(x)| ~\leq ~ \lceil x\rceil\left|{\textstyle{{{x-1}\choose{p}}}}\right|\left|\Delta^pg(n)\right|,\qquad x>0,~n\in I\cap\mathbb{N}^*. \end{aligned}$$

  If p ≥ 1, we also have the following tighter inequality

  $$\displaystyle \begin{aligned} |f^p_n[g](x)-f(x)| ~\leq ~ \left|{\textstyle{{{x-1}\choose{p}}}}\right|\left|\Delta^{p-1}g(n+x)-\Delta^{p-1}g(n)\right|,\quad x>0,~n\in I\cap\mathbb{N}^*. \end{aligned}$$

• (c) The sequence $n\mapsto f^p_n[g]$ converges uniformly on any bounded subset of $\mathbb {R}_+$ to f.

Proof

We have that $g\in \mathcal {D}^p_{\mathrm {S}}\subset \mathcal {D}^{p+1}_{\mathrm {S}}$. By Lemma 2.7, it follows immediately that g lies in $\mathcal {R}^{p+1}_{\mathrm {S}}$, and hence also in $\mathcal {R}^p_{\mathrm {S}}$ by (3.4) and (3.5). This establishes assertion (a). Now, suppose for instance that g lies in $\mathcal {K}^p_+$. Let I be any unbounded subinterval of $\mathbb {R}_+$ on which g is p-convex and let $m\in I\cap \mathbb {N}^*$. For any x > 0, the sequence $k\mapsto \rho ^{p+1}_k[g](x)$ for k ≥ m does not change in sign by Lemma 2.7. Thus, since g lies in $\mathcal {D}^p_{\mathbb {N}}$ , for any x > 0 the series

$$\displaystyle \begin{aligned} \sum_{k=m}^{\infty}\rho^{p+1}_k[g](x) \end{aligned}$$

converges by Lemma 3.5. By (3.8) it follows that the sequence $n\mapsto f^p_n[g](x)$ converges. Denoting the limiting function by f, we necessarily have f(1) = 0. Moreover, by (3.7) and assertion (a) we must have Δf = g.

Since g is p-convex on I, for every $n\in \mathbb {N}^*$ the function $f^p_n[g]$ is clearly p-concave on I. (Note that the second sum in (1.4) is a polynomial of degree less than or equal to p in x, hence by Proposition 2.4 it is both p-convex and p-concave.) Since f is a pointwise limit of functions p-concave on I, it too is p-concave on I.

The claimed inequalities then follow from identity (3.3), Lemma 2.7, and the observation that the restriction of the sequence n↦ Δ^p g(n) to $I\cap \mathbb {N}^*$ increases to zero by Lemma 2.5. Indeed, for any x > 0 and any $n\in I\cap \mathbb {N}^*$, we then have

$$\displaystyle \begin{aligned} \begin{array}{rcl} |f^p_n[g](x)-f(x)| & =&\displaystyle |\rho^{p+1}_n[f](x)| ~\leq ~ \left|{\textstyle{{{x-1}\choose{p}}}}\right|\left|\Delta^pf(n+x)-\Delta^pf(n)\right|\\ & \leq &\displaystyle \left|{\textstyle{{{x-1}\choose{p}}}}\right|\,\sum_{j=0}^{\lceil x\rceil -1}|\Delta^p g(j+n)| ~\leq ~ \lceil x\rceil\left|{\textstyle{{{x-1}\choose{p}}}}\right|\left|\Delta^pg(n)\right|. \end{array} \end{aligned} $$

This proves assertion (b). Assertion (c) immediately follows from the first inequality of assertion (b). □

Remark 3.7

We have shown in Theorems 3.1 and 3.6 that the sequence $n\mapsto f_n^p[g]$ converges uniformly on any bounded subset of $\mathbb {R}_+$. In fact, we have proved the slightly more general property that the sequence $n\mapsto \rho _n^{p+1}[f]$ converges uniformly on any bounded subset of [0, ∞) to 0. $\lozenge$

Theorems 3.1 and 3.6 show that the assumption that $g\in \mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$ constitutes a sufficient condition to ensure both the uniqueness (up to an additive constant) and existence of solutions to the equation Δf = g that lie in $\mathcal {K}^p$. Nevertheless, we can show that this condition is actually not quite necessary. We discuss and elaborate on this natural question in Appendix C.

We now present an important property of the sequence $n\mapsto f^p_n[g]$. Considering the straightforward identity

$$\displaystyle \begin{aligned} f^{p+1}_n[g](x)-f^p_n[g](x) ~=~ {\textstyle{{{x}\choose{p+1}}}}\,\Delta^pg(n), \end{aligned}$$

we immediately see that if the sequence

$$\displaystyle \begin{aligned} n ~\mapsto ~f^{p+1}_n[g](x)-f^p_n[g](x) \end{aligned}$$

approaches zero for some $x\in \mathbb {R}_+\setminus \{0,1,\ldots ,p\}$, then g must lie in $\mathcal {D}^p_{\mathbb {N}}$. More importantly, the identity above also shows that if g lies in $\mathcal {D}^p_{\mathbb {N}}$ and if the sequence $n\mapsto f^p_n[g](x)$ converges, then so does the sequence $n\mapsto f^{p+1}_n[g](x)$ and both sequences converge to the same limit. Since the inclusion $\mathcal {D}^p_{\mathbb {N}}\subset \mathcal {D}^{p+1}_{\mathbb {N}}$ holds for any $p\in \mathbb {N}$, we immediately obtain the following important proposition.

Proposition 3.8

Let $p\in \mathbb {N}$ . If $g\in \mathcal {D}^p_{\mathbb {N}}$ and if the sequence $n\mapsto f^p_n[g](x)$ converges, then for any integer q ≥ p the sequence

$$\displaystyle \begin{aligned} n ~\mapsto ~|f^p_n[g](x)-f^q_n[g](x)| \end{aligned}$$

converges to zero. Moreover, the convergence is uniform on any bounded subset of $\mathbb {R}_+$.

Let us end this section with the following observation about our uniqueness and existence results. In Theorem 3.1, we have proved the uniqueness of the solution f that lies in $\mathcal {K}^p$ by first proving that this solution necessarily lies in $\mathcal {R}^{p+1}_{\mathrm {S}}$. Although this latter asymptotic condition may seem a bit less natural than the assumption that f lies in $\mathcal {K}^p$, we could as well consider it as a sufficient condition to guarantee uniqueness. A similar observation can be made for the existence Theorem 3.6. We can therefore establish the following two alternative results.

Proposition 3.9 (Uniqueness)

Let $p\in \mathbb {N}$ and let $g\colon \mathbb {R}_+\to \mathbb {R}$ . Suppose that $f\colon \mathbb {R}_+\to \mathbb {R}$ is a solution to the equation Δf = g that lies in $\mathcal {R}^{p+1}_{\mathbb {N}}$ . Then assertion (b) of Theorem 3.1 holds, and hence f is unique (up to an additive constant).

Proof

This follows immediately from identity (3.3). □

Proposition 3.10 (Existence)

Let $p\in \mathbb {N}$ and suppose that the function $g\colon \mathbb {R}_+\to \mathbb {R}$ lies in $\mathcal {D}^p_{\mathbb {N}}$ and has the property that, for each x > 0, the sequence $n\mapsto \rho ^{p+1}_n[g](x)$ is summable. Then g lies in $\mathcal {R}_{\mathbb {N}}^p$ and there exists a unique (up to an additive constant) solution $f\colon \mathbb {R}_+\to \mathbb {R}$ to the equation Δf = g that lies in $\mathcal {R}^{p+1}_{\mathbb {N}}$.

Proof

Since the sequence $n\mapsto \rho ^{p+1}_n[g](x)$ is summable, by (3.8) the sequence $n\mapsto f^p_n[g](x)$ converges. Denoting the limiting function by f, we necessarily have f(1) = 0. By (3.6), the function g necessarily lies in $\mathcal {R}_{\mathbb {N}}^{p+1}$, and hence also in $\mathcal {R}_{\mathbb {N}}^p$ by (3.4) and (3.5). Thus, we must have Δf = g by (3.7) and f lies in $\mathcal {R}^{p+1}_{\mathbb {N}}$ by (3.3). □

Example 3.11

Let us apply Proposition 3.9 to $g(x)=\ln x$ and p = 1. We then obtain the following alternative characterization of the gamma function (in the multiplicative notation).

If $f\colon \mathbb {R}_+\to \mathbb {R}_+$ is a solution to the equation f(x + 1) = x f(x) having the asymptotic property that, for each x > 0,

$$\displaystyle \begin{aligned} f(x+n) ~\sim ~ n^x f(n)\qquad \mbox{as }n\to_{\mathbb{N}}\infty, \end{aligned}$$

then f(x) = c  Γ(x) for some c > 0.

It is easy to see that this characterization also holds on the whole complex domain of the gamma function, namely $\mathbb {C}\setminus (-\mathbb {N})$. $\lozenge$

3.2 The Case when the Sequence g(n) Is Summable

Let $\mathcal {D}^{-1}_{\mathbb {N}}$ be the set of functions $g\colon \mathbb {R}_+\to \mathbb {R}$ having the asymptotic property that the series $\sum _{k=1}^{\infty }g(k)$ converges. We immediately observe that $\mathcal {D}^{-1}_{\mathbb {N}}\subset \mathcal {D}^0_{\mathbb {N}}$. In this context, our uniqueness and existence results can be complemented by the following two theorems.

Theorem 3.12 (Uniqueness)

Let $g\in \mathcal {D}^{-1}_{\mathbb {N}}$ and suppose that $f\colon \mathbb {R}_+\to \mathbb {R}$ is a solution to the equation Δf = g that lies in $\mathcal {K}^0$ . Then, the following assertions hold.

• (a) f(x) has a finite limit as x →∞, denote it by f(∞).

• (b) For each x > 0, the series $\sum _{k=0}^{\infty }g(x+k)$ converges and we have

  $$\displaystyle \begin{aligned} f(x) ~=~ f(\infty) - \sum_{k=0}^{\infty}g(x+k){\,}. \end{aligned}$$

• (c) The series $x\mapsto \sum _{k=0}^{\infty }g(x+k)$ converges uniformly on $\mathbb {R}_+$ to f(∞) − f.

Proof

The sequence n↦f(n) converges by (3.1). Assuming for instance that f lies in $\mathcal {K}^0_+$, for any x > 0 we obtain

$$\displaystyle \begin{aligned} f(\lfloor x\rfloor +n) ~\leq ~ f(x+n) ~\leq ~ f(\lceil x\rceil +n)\quad \mbox{for large integer }n. \end{aligned}$$

Letting $n\to _{\mathbb {N}}\infty$ in these inequalities and using the squeeze theorem, we get assertion (a). Assertion (b) follows from assertion (a) and identity (3.2). Now, for large integer n, by assertion (b) and identity (3.2) we have

$$\displaystyle \begin{aligned} \sup_{x\in\mathbb{R}_+}\left|\sum_{k=n}^{\infty}g(x+k)\right| ~=~ \sup_{x\in\mathbb{R}_+}\left|f(x+n)-f(\infty)\right| ~\leq ~ |f(n)-f(\infty)|. \end{aligned}$$

This proves assertion (c). □

Theorem 3.13 (Existence)

Let $g\in \mathcal {D}^{-1}_{\mathbb {N}}\cap \mathcal {K}^0$ . The following assertions hold.

• (a) We have that $g\in \mathcal {R}^0_{\mathbb {N}}$.

• (b) The series $\sum _{k=0}^{\infty }g(x+k)$ converges for every x > 0, and the function $f\colon \mathbb {R}_+\to \mathbb {R}$ defined by

  $$\displaystyle \begin{aligned} f(x) ~=~ -\sum_{k=0}^{\infty}g(x+k){\,},\qquad x>0, \end{aligned} $$

  (3.9)

  is a solution to the equation Δf = g that is decreasing (resp. increasing) on any unbounded subinterval I of $\mathbb {R}_+$ on which g is increasing (resp. decreasing). Moreover, we have f(x) → 0 as x →∞ and, for every $n\in I\cap \mathbb {N}^*$,

  $$\displaystyle \begin{aligned} \left|\sum_{k=n}^{\infty}g(x+k)\right| ~=~ |f(x+n)| ~\leq ~ |f(n)|,\qquad x>0. \end{aligned}$$

• (c) The series $x\mapsto \sum _{k=0}^{\infty }g(x+k)$ converges uniformly on $\mathbb {R}_+$ to − f.

Proof

By Theorem 3.6, assertion (a) clearly holds (since g also lies in $\mathcal {D}^0_{\mathbb {N}}$) and, for each x > 0, the series (3.9) converges and is a solution to the equation Δf = g that satisfies the claimed monotonicity properties. Theorem 3.12 then shows that the function f vanishes at infinity. The rest of assertion (b) follows from (3.2). Assertion (c) is then immediate. □

Theorems 3.12 and 3.13 motivate the following definition.

Definition 3.14

For any $\mathrm {S}\in \{\mathbb {N},\mathbb {R}\}$, we let $\mathcal {D}^{-1}_{\mathrm {S}}$ denote the set of functions $g\colon \mathbb {R}_+\to \mathbb {R}$ having the asymptotic property that, for each x ∈S, the series

$$\displaystyle \begin{aligned} \sum_{k=0}^{\infty}g(x+k) \end{aligned}$$

converges and tends to zero as x →_S ∞.

Clearly, this definition is consistent with our prior definition of $\mathcal {D}^{-1}_{\mathbb {N}}$ and we can immediately see that the inclusion $\mathcal {D}^{-1}_{\mathbb {R}}\subset \mathcal {D}^{-1}_{\mathbb {N}}$ holds. Moreover, by Theorem 3.13 we have that

$$\displaystyle \begin{aligned} \mathcal{D}^{-1}_{\mathbb{R}}\cap\mathcal{K}^0 ~=~ \mathcal{D}^{-1}_{\mathbb{N}}\cap\mathcal{K}^0. \end{aligned} $$

(3.10)

Example 3.15 (The Trigamma Function)

The trigamma function ψ ₁ is defined on $\mathbb {R}_+$ as the derivative ψ′ of the digamma function. Hence, it has the property that

$$\displaystyle \begin{aligned} \Delta\psi_1(x) ~=~ D\Delta\psi(x) ~=~ -1/x^2\qquad \mbox{for all }x>0. \end{aligned}$$

Since the function ψ lies in $\mathcal {D}^1_{\mathbb {N}}\cap \mathcal {K}^1_-$, one can show (see Proposition 4.12 in the next chapter) that ψ ₁ lies in $\mathcal {D}^0_{\mathbb {N}}\cap \mathcal {K}^0_-$. Now, the function g(x) = −1∕x ² clearly lies in $\mathcal {D}^{-1}_{\mathbb {N}}\cap \mathcal {K}^0_+$ and hence also in $\mathcal {D}^0_{\mathbb {N}}\cap \mathcal {K}^0_+$. It also lies in $\mathcal {D}^{-1}_{\mathbb {R}}\cap \mathcal {K}^0_+$ by (3.10). Thus, by Theorems 3.6, 3.12, and 3.13, we see that the trigamma function ψ ₁ is the unique decreasing solution f to the equation Δf = g that vanishes at infinity. Moreover, we have that

$$\displaystyle \begin{aligned} \psi_1(x) ~=~ \sum_{k=0}^{\infty}\frac{1}{(x+k)^2}\qquad \mbox{and}\qquad \psi_1(1) ~=~ \sum_{k=1}^{\infty}\frac{1}{k^2} ~=~ \frac{\pi^2}{6}{\,}. \end{aligned}$$

Furthermore, the sequence of functions

$$\displaystyle \begin{aligned} n ~\mapsto ~ \sum_{k=0}^{n-1}\frac{1}{(x+k)^2} ~=~ \psi_1(x)-\psi_1(x+n) \end{aligned}$$

converges uniformly on $\mathbb {R}_+$ to the function ψ ₁(x), and Theorem 3.13 provides the following inequalities

$$\displaystyle \begin{aligned} 0 ~ \leq ~ \psi_1(x+n) ~=~ \sum_{k=n}^{\infty}\frac{1}{(x+k)^2} ~\leq ~ \psi_1(n){\,},\qquad x>0,~n\in\mathbb{N}^*. \end{aligned}$$

Finally, Theorem 3.6 provides the following additional inequalities

$$\displaystyle \begin{aligned} 0 ~ \leq ~ \psi_1(n)-\psi_1(x+n) ~\leq ~ \frac{\lceil x\rceil}{n^2}{\,},\qquad x>0,~n\in\mathbb{N}^*. \end{aligned}$$

We will further investigate the trigamma function ψ ₁ as a special polygamma function in Sect. 10.3. $\lozenge$

3.3 Historical Notes

As mentioned in Chap. 1, the uniqueness and existence result in the case when p = 1 was established in the pioneering work of Krull [54, 55] and then independently by Webster [97, 98] as a generalization of Bohr-Mollerup’s theorem . We observe that it was also partially rediscovered by Dinghas [33]. In addition, we note that Krull’s result was presented and somewhat revisited by Kuczma [56] (see also Kuczma [59] and Kuczma [60, pp. 114–118]) as well as by Anastassiadis [7, pp. 69–73]. To our knowledge, the only attempts to establish uniqueness and existence results for any value of p were made by Kuczma [60, pp. 118–121] and Ardjomande [9]. Independently of these latter results, an investigation of the special case when p = 2, illustrated by the Barnes G-function, was made by Rassias and Trif [86] (see our Appendix B).

We also observe that Gronau and Matkowski [44, 45] improved the multiplicative version of Krull’s result by replacing the log-convexity property with the much weaker condition of geometrical convexity (see also Guan [46] for a recent application of this result), thus providing another characterization of the gamma function, which was later improved by Alzer and Matkowski [4] and Matkowski [68, 69]. (For further characterizations of the gamma function and generalizations, see also Anastassiadis [7] and Muldoon [79].)

Many other variants and improvements of Krull’s result can actually be found in the literature. For instance, Anastassiadis [6] (see also Anastassiadis [7, p. 71]) generalized it by modifying the asymptotic condition. Rohde [88] also generalized it by modifying the convexity property. Gronau [42] proposed a variant of Krull’s result and applied it to characterize the Euler beta and gamma functions and study certain spirals (see also Gronau [43]). Merkle and Ribeiro Merkle [71] proposed to combine Krull’s approach with differentiation techniques to characterize the Barnes G-function. Himmel and Matkowski [48] also proposed improvements of Krull’s result to characterize the beta and gamma functions.