\documentclass[12pt]{article} \setlength{\textwidth}{6.3in} \setlength{\textheight}{8.7in} \setlength{\topmargin}{0pt} \setlength{\headsep}{0pt} \setlength{\headheight}{0pt} \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt} \makeatletter \newfont{\footsc}{cmcsc10 at 8truept} \newfont{\footbf}{cmbx10 at 8truept} \newfont{\footrm}{cmr10 at 10truept} \renewcommand{\ps@plain}{% \renewcommand{\@oddfoot}{\footsc the electronic journal of combinatorics {\footbf 11(2)} (2005), \#A2\hfil\footrm\thepage}} \makeatother \pagestyle{plain} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \def\C{\mathbb{C}} \def\Q{\mathbb{Q}} \def\Z{\mathbb{Z}} \def\Li{\operatorname{Li}} \newcommand{\EIS}[1]{\emph{EIS}~#1} \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newcounter{noteno} \newenvironment{note}{\refstepcounter{noteno}% \medbreak\noindent{\bf Note~\thenoteno.}}% {\hfill{\Large $\lhd$}\medbreak} \begin{document} \title{On the non-holonomic character of logarithms, powers, and the $n$th prime function} \author{Philippe Flajolet\\ \small Algorithms Project,\\[-0.8ex] \small INRIA Rocquencourt, F-78153 Le Chesnay (France)\\[-0.8ex] \small \texttt{Philippe.Flajolet AT inria.fr} \and Stefan Gerhold\thanks{Supported in part by the SFB-grant F1305 of the Austrian FWF}\\ \small Research Institute for Symbolic Computation,\\[-0.8ex] \small Johannes Kepler University Linz (Austria)\\[-0.8ex] \small \texttt{stefan.gerhold AT risc.uni-linz.ac.at} \and Bruno Salvy\\ \small Algorithms Project,\\[-0.8ex] \small INRIA Rocquencourt, F-78153 Le Chesnay (France)\\[-0.8ex] \small \texttt{Bruno.Salvy AT inria.fr}} \date{\small Submitted: Jan 21, 2005; Accepted: Mar 30, 2005; Published: Apr 28, 2005 \\ \small Mathematics Subject Classifications: 05A15, 11B83, 33E30} \maketitle \begin{small} \begin{flushright} \sl Es ist eine Tatsache, da{\ss} die genauere Kenntnis\\ des Verhaltens einer analytischen Funktion\\ in der N{\"a}he ihrer singul\"aren Stellen\\ eine Quelle von arithmetischen S{\"a}tzen ist.\footnote{% \em ``It is a fact that the precise knowledge of the behaviour of an analytic function in the vicinity of its singular points is a source of arithmetic properties.'' }\\[2mm] {\sc --- Erich Hecke} \rm\cite[Kap.~VIII]{Hecke23} \end{flushright} \end{small} \begin{abstract} We establish that the sequences formed by logarithms and by ``fractional'' powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [\emph{El. J. Comb.} {\bf 11} (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularity-based methods and several naturally occurring sequences are proved to be non-holonomic. \end{abstract} \section*{Introduction} A sequence $(f_n)_{n\ge0}$ of complex numbers is said to be \emph{holonomic} (or \emph{$P$-recursive}) if it satisfies a linear recurrence with coefficients that are polynomial in the index~$n$, that is, \begin{equation}\label{h1} p_0(n)f_{n+d} +p_1(n)f_{n+d-1}+\cdots+p_d(n)f_{n}=0, \qquad n\ge 0, \end{equation} for some polynomials $p_j(X)\in\C[X]$. A formal power series $f(z)=\sum_{n\ge0}f_n z^n$ is \emph{holonomic} (or \emph{$\partial$-finite}) if it satisfies a linear differential equation with coefficients that are polynomial in the variable~$z$, that is, \begin{equation}\label{h2} q_0(z)\frac{d^e}{dz^e} f(z)+q_1(z)\frac{d^{e-1}}{dz^{e-1}} f(z)+\cdots +q_e(z) f(z)=0, \end{equation} for some polynomials $q_k(X)\in\C[X]$. It is well known that a sequence is holonomic if and only if its generating series is holonomic. (See Stanley's book~\cite{Stanley98} for the basic properties of these sequences and series.) By extension, a function analytic at~0 is called holonomic if its power series representation is itself holonomic. Holonomic sequences encapsulate many of the combinatorial sequences of common interest, for instance, a wide class of sums involving binomial coefficients. At the same time, they enjoy a varied set of closure properties and several formal mechanisms have been recognised to lead systematically to holonomic sequences---what we have in mind here includes finite state models and regular grammars leading to rational (hence, holonomic) functions, context-free specifications leading to algebraic (hence, holonomic) functions, the multivariate holonomic framework initiated by Lipshitz and Zeilberger~\cite{Lipshitz88,Lipshitz89,Zeilberger90}, a wide class of problems endowed with symmetry conducive to holonomic functions (via Gessel's theory~\cite{Gessel90}). On these aspects, we may rely on general references like~\cite{FlSe04,Stanley98} as well as on many works of Zeilberger, who is to be held accountable for unearthing the power of the holonomic framework accross combinatorics; see~\cite{PeWiZe96,Zeilberger90}. Thus, in a way, a non-holonomicity result represents some sort of a structural complexity lower bound. This note answers three problems described as open in an article of Stefan Gerhold~\cite{Gerhold04} very recently published in the \emph{Electronic Journal of Combinatorics}. \begin{proposition}\label{log-thm} The sequence $f_n=\log n$ is not holonomic. \end{proposition} \noindent (For definiteness, we agree that $\log 0 \equiv 0$ here.) For~$\alpha$ an integer, the sequence \[ h_n=n^\alpha\] is clearly holonomic. (As a matter of fact, the generating function is rational if $\alpha\in\Z_{\ge0}$ and of polylogarithmic type if $\alpha\in\Z_{<0}$.) Gerhold~\cite{Gerhold04} proved that for any $\alpha$ that is rational but not integral, $h_n$ fails to be holonomic. For instance, $h_n=\sqrt{n}$ fails to be holonomic because, in essence, $\Q(\sqrt{2},\sqrt{3},\ldots)$ is not a finite extension of~$\Q$. \begin{proposition}\label{pow-thm} For $\alpha\in\C$, the sequence of powers $h_n=n^\alpha$ is holonomic if and only if $\alpha\in\Z$. \end{proposition} \noindent (We agree that $h_0=0$.) \begin{proposition}\label{prime-thm} The sequence $g_n$ defined by the fact that $g_n$ is the $n$th prime is non-holonomic. \end{proposition} \noindent (We agree that $g_0=1$, $g_1=2$, $g_2=3$, $g_3=5$, and so on.) \smallskip Proposition~\ref{log-thm}, conjectured by Gerhold in~\cite{Gerhold04} was only proved under the assumption that a difficult conjecture of number theory (Schanuel's conjecture) holds. The author of~\cite{Gerhold04} describes the statement of our Proposition~\ref{pow-thm} as a \emph{``natural conjecture''}. Proposition~\ref{prime-thm} answers an explicit question of Gerhold who writes: \emph{``we do not know of any proof that the sequence of primes is non-holonomic''.} \smallskip Our proofs are plainly based on the combination of two facts. First, holonomic objects satisfy rich closure properties. In particular, we make use of closure of under sum, product and composition with an algebraic function. Second, the asymptotic behaviour of holonomic sequences, which is reflected by the asymptotic behaviour at singularities of their generating functions, is rather strongly constrained. For instance, iterated logarithms or negative powers of logarithms are ``forbidden'' and estimates like \begin{equation}\label{e0} a_n \mathop{\sim}_{n\to+\infty} \log\log n, \qquad b(z)\mathop{\sim}_{z\to1-} \frac{1}{\log(1-z)}, \end{equation} are sufficient to conclude that the sequence $(a_n)$ and the function $b(z)$ are non-holonomic. (See details below.) A conjunction of the previous two ideas then perfectly describes the strategy of this note: In order to prove that a sequence is non-holonomic, it suffices to locate a ``derived sequence'' (produced by holonomicity-preserving transformations) that exhibits a suitable combination of kosher asymptotic terms with a foreign non-holonomic element, like in~(\ref{e0}). We choose here to operate directly with generating functions. Under this scenario, one can rely on the well established classification of singularities elaborated at the end of the nineteenth century by Fuchs~\cite{Fuchs66}, Fabry~\cite{Fabry85} and others. A summary of what is known is found in standard treatises, for instance the ones by Wasow~\cite{Wasow87} and Henrici~\cite{Henrici77}. What we need of this theory is summarised by Theorem~\ref{classif-thm} of the next section. The relation between asymptotic behaviour of sequences and local behaviour of the generating functions is provided by a classical Abelian theorem, stated as Theorem~\ref{abel-thm} below. \smallskip \begin{note} A methodological remark is in order at this stage. A glance at~(\ref{e0}) suggests two possible paths for proving non-holonomicity: one may \emph{a priori} operate equally well with sequences or with generating functions. The latter is what we have opted to do here. The former approach with sequences seems workable, but it requires a strong structure theorem analogous to Theorem~\ref{classif-thm} below for recurrences, i.e., difference equations. An ambitious programme towards such a goal was undertaken by Birkhoff and Trjitzinsky~\cite{Birkhoff38,BiTr30} in the 1930's, their works being later followed by Wimp and Zeilberger in~\cite{WiZe85}. However, what is available in the classical literature is largely a set of \emph{formal} solutions to difference equations and recurrences, and the relation of these to actual (analytic) solutions represents a difficult problem evoked in~\cite[p.~168]{WiZe85} and~\cite[p.~1138]{Odlyzko95}; see also~\cite{BrFaIm00} for recent results relying on multisummability. \end{note} \begin{note} In this short paper, we do nothing but assemble some rather well-known facts of complex analysis, and logically organise them towards the goal of proving certain sequences to be non-holonomic. Our purpose is thus essentially pedagogical. As it should become transparent soon, a rough heuristic in this range of problem is the following: \emph{Almost anything is non-holonomic unless it is holonomic by design}. (This na{\"\i}ve remark cannot of course be universally true and there are surprises, e.g., some sequences may eventually admit algebraic or holonomic descriptions for rather deep reasons. Amongst such cases, we count the enumeration of~$k$-regular graphs and various types of maps~\cite{Gessel90,GoJa83}, the enumeration of permutations with bounded-length increasing subsequences, the Ap\'ery sequence~\cite{Poorten79} related to a continued fraction expansion of~$\zeta(3)$, as well as the appearance of holonomic functions in the theory of modular forms, for which we refer to the beautiful exposition of Kontsevich-Zagier~\cite{KoZa01}.) \end{note} \section{Methods} From the most basic theorems regarding the existence of analytic solutions to differential equations (e.g., \cite[Th.~9.1]{Henrici77}), any function~$f(z)$ analytic at~0 that is holonomic can be continued analytically along any path that avoids the finite set~$\Sigma$ of points defined as roots of the equation $p_0(z)=0$, where $p_0$ is the leading coefficient in~(\ref{h1}). Figuratively: \begin{theorem}[Finiteness of singularities]\label{finite-thm} A holonomic function has only finitely many singularities. \end{theorem} This theorem gives immediately as non-holonomic a number of sequences enumerating classical combinatorial structures. \begin{itemize} \item[---] Integer partitions, whose generating function is $P(z)=\prod(1-z^n)^{-1}$, as the function admits the unit circle as a natural boundary. The same argument applies to integer partitions with summands restricted to any infinite set (e.g., primes), partitions into distict summands, plane partitions, and so on. More generally, combinatorial classes defined by an unlabelled set or multiset construction~\cite{FlSe04} are non-holonomic, unless a rather drastic combinatorial simplification occurs. \item[---] Alternating (also known as zig-zag, up-and-down, cf\footnote{% In order to keep this note finite, we refer to some of the combinatorial problems by means of their number in Sloane's \emph{Encyclopedia of Integer Sequences (EIS)}, see~\cite{Sloane00}. }~\EIS{A000111}) permutations with exponential generating function $\tan z + \sec z$, as they have the odd multiples of $\frac\pi2$ as set of poles\footnote{% Stanley~\cite{Stanley80} describes an algebraic proof dependent on the fact that $\exp(z)$ is nonalgebraic (his Example~4.5), then goes on to observe in his~\S4.a that $\sec z$ ``\emph{has infinitely many poles}''.}. A similar argument applies to preferential arrangements (also known as ordered set partitions or surjections, cf \EIS{A000670}), Bernoulli numbers, and the like. \item[---] Necklaces (equivalently Lyndon words, irreducible polynomials over finite fields), whose generating function admits the unit circle as a natural boundary. More generally, ``most'' unlabelled cycles are non-holonomic. \item[---] Unlabelled plane trees (\EIS{A000081}), whose implicit specification involves an unlabelled multiset construction. \item[---] Knight's walks in the quarter plane (\EIS{A057790}). This is described by a simple two-dimensional linear recurrence with constant coefficients. Bousquet-M\'elou and Petkov\v sek~\cite{BoPe00} have established that an associated generating function has infinitely many singularities. \end{itemize} In many cases, the criterion above is too brutal. For instance it does not preclude holonomicity for the Cayley tree function, \begin{equation}\label{cayley} T(z)=\sum_{n\ge1} n^{n-1}\frac{z^n}{n!}. \end{equation} Indeed, the (multivalued) function~$T(z)$ has singularities at $0,\infty,e^{-1}$ only. \smallskip A major theorem constrains the possible growth of a holonomic function near any of its singularities. Paraphrasing Theorem~19.1 of~\cite[p.~111]{Wasow87}, we can state: \begin{theorem}[Structure Theorem for singularities]\label{classif-thm} Let there be given a differential equation of the form~\eqref{h2}, a singular point $z_0$, and a sector~$S$ with vertex at~$z_0$. Then, for $z$ in a sufficiently narrow subsector~$S'$ of~$S$ and for $|z-z_0|$ sufficiently small, there exists a basis of~$d$ linearly independent solutions to~\eqref{h2}, such that any solution~$Y$ in that basis admits as $z\to z_0$ in the subsector an asymptotic expansion of the form \begin{equation}\label{struct-eqn} Y\sim \exp\left(P(Z^{-1/r})\right)z^\alpha \sum_{j=0}^\infty Q_j(\log Z)Z^{js}, \qquad Z:=(z-z_0), \end{equation} where $P$ is a polynomial, $r$ an integer of~$\Z_{\ge0}$, $\alpha$ a complex number, $s$ a rational of $\Q_{>0}$, and the $Q_j$ are a family of polynomials of uniformly bounded degree. The quantities $r,P,\alpha,s,Q_j$ depend on the particular solution and the formal asymptotic expansions of~\eqref{struct-eqn} are $\C$-linearly independent. % but two % different elements of the basis cannot have identical quantities % $r,P,s,Q_j$. \end{theorem} (The argument is based on first constructing a \emph{formal} basis of independent solutions, each of the form~(\ref{struct-eqn}), and then applying to the possibly divergent expansions a summation mechanism that converts such formal solutions into actual analytic solutions. The restriction of the statement to a subsector is related to the Stokes phenomena associated to so-called ``irregular'' singularities.) \smallskip This theorem implies that the sequence $(n^{n-1}/n!)$ (hence $(n^n)$) is non-holonomic. Indeed the Cayley tree function satisfies the functional equation \[ T(z)=ze^{T(z)}, \] corresponding to the fact that it enumerates labelled nonplane trees. Set $W(z)=-T(-z)$, which is otherwise known as the ``Lambert W-function''. One has \[ W(x)\mathop{=}_{x\to+\infty} \log x - \log\log x +O(1) ,\] as verified by bootstrapping (see De Bruijn's monograph~\cite[p.~26]{deBruijn81}). This is enough to conclude that $W$, hence $T$, is non-holonomic as the $\log\log(\cdot)$ term is incompatible with Eq.~(\ref{struct-eqn}). Observe that, conceptually, the proof involves considering the \emph{analytic continuation} of $T(z)$ and then extracting a clearly non-holonomic term in the expansion near a singularity. More of this in the next sections. Amongst other applications, we may cite: \begin{itemize} \item[---] Stanley's children rounds [\EIS{A066166}], with exponential generating function $(1-z)^{-z}$. The expansion as $z\to1$, \[ (1-z)^{-z} \mathop{\sim}_{z\to1} \frac{1}{1-z}\left(1+(1-z)\log(1-z) +\frac{(1-z)^2\log^2(1-z)}{2!}+\cdots\right), \] contradicts the fact that logarithms can only appear with bounded degrees in holonomic functions. \item[---] Bell numbers have OGF $e^{e^z-1}$. In this case, the double exponential behaviour as $z\to+\infty$ excludes them from the holonomic ring. \end{itemize} \smallskip Finally, what is given is often a sequence rather than a function. Under such circumstances, it proves handy to be able to relate the asymptotic behaviour of~$f_n$ as $n\to+\infty$ to the asymptotic form of its generating function $f(z)$, near a singularity. Such transfers exist and are widely known in the literature as \emph{Abelian theorems}. We make use here of well-established principles in this theory, as found, e.g., in the reference book by Bingham, Goldie, and Teugels~\cite{BiGoTe89}. For convenience of exposition, we state explicitly one version used repeatedly here: \begin{theorem}[Basic Abelian theorem]\label{abel-thm} Let $\phi(x)$ be any of the functions \begin{equation}\label{condabel} x^\alpha (\log x)^{\beta}(\log\log x)^\gamma,\qquad\alpha\ge0, \quad \beta,\gamma\in\C. \end{equation} Let $(u_n)$ be a sequence that satisfies the asymptotic estimate \[ u_n \mathop{\sim}_{n\to\infty} \phi(n). \] Then, the generating function, \[ u(z):=\sum_{n\ge 0} u_n z^n, \] satisfies the asymptotic estimate \begin{equation}\label{estim} u(z)\mathop{\sim}_{z\to1-} \Gamma(\alpha+1)\frac{1}{(1-z)}\phi\left(\frac{1}{1-z}\right). \end{equation} This estimate remains valid when $z$ tends to~1 in any sector with vertex at~$1$, symmetric about the horizontal axis, and with opening angle $<\pi$. \end{theorem} \begin{proof}[Proof (sketch)] We shall content ourselves here with brief indications since Corollary 1.7.3 p.~40 of~\cite{BiGoTe89} provides simultaneously the needed Abelian property and its real-analysis Tauberian converse\footnote{% The singularity analysis technology of Flajolet and Odlyzko~\cite{FlOd90b,Odlyzko95} provides sufficient conditions for the converse complex-Tauberian implication.}, at least in the case when $z$ tends to~$1^-$ along the real axis. For simplicity, consider first the representative case where $\phi(x)=\log\log x$ and one has exactly $u_n=\phi(n)$ for $n\ge2$, with $u_0=u_1=0$. Assume at this stage that~$z$ is real positive and set $z=e^{-t}$, where $t\to0$ as $z\to1$. We have \[ u(z) = \sum_{n\ge2} \phi(n) e^{-nt}. \] Take $n_1=\lfloor t^{-1}/\log t^{-1}\rfloor$. Basic majorizations imply that the sum of the terms corresponding to $nn_2$ is easily checked to be $O(1)$. The remaining ``central'' terms $n_1\le n\le n_2$ are such that $\phi(n)$ varies slowly over the interval and one has $\phi(n_1)\sim\phi(n_2)\sim \phi(1/t)$. One can thus take out a factor of $\phi(1/t)$ and conclude, upon approximating the sum by an integral, that \begin{equation}\label{abelard} \sum_{n=n_1}^{n_2} \phi(n) e^{-nt}\sim \frac{\phi(1/t)}{t}\int_{1/\log t^{-1}}^{\log t^{-1}} e^{-x}\, dx \sim \frac{\log\left(\log(1-z)^{-1}\right)}{1-z}. \end{equation} (Use the Euler-Maclaurin summation formula, then complete the tails.) The proof above applies when \[ z=e^{-t+i\theta},\qquad \hbox{with}\quad |\theta|<\theta_0,\] for some $\theta_0<\frac{\pi}{2}$. Once more only the central terms matter asymptotically; the integral is then to be taken along a line of angle $\theta$, but it reduces to the corresponding integral along the positive real line, by virtue of the residue theorem. This suffices to justify the extension of the estimate to sectors. The case $\phi(x)=\log\log x$ is then settled. The extension to $u_n=\phi(n)$ when $\phi$ only involves powers of logarithms and of iterated logarithms follows similar lines, as $\log n$ is also of slow variation. The inclusion of a power of $n$, in the form $n^{\alpha}$ implies that the integral in~(\ref{abelard}) should be modified to include a factor $t^{\alpha}$ leading to a real line integral that evaluates to the Gamma function, $\Gamma(\alpha+1)$. Finally, simple modifications of the previous arguments show that if a sequence $(v_n)$ satisfies $v_n=o(\phi(n))$, for $\phi(n)$ any of the sequences of~(\ref{condabel}), then its generating function~$v(z)$ is a little-oh of the right hand side of~(\ref{estim}). Decomposing $u_n=\phi(n)+v_n$ completes the proof of the statement. \end{proof} \begin{note} We have chosen to state the Abelian Theorem (Theorem~\ref{abel-thm}) for $z$ varying in a cone of the complex plane, rather than the more customary real line. In this way we can avail ourselves of the comparatively simple Structure Theorem (as stated above in Theorem~\ref{classif-thm}) and avoid some of the possible hardships due to the Stokes phenomenon. \end{note} Here is a direct application of Theorem~\ref{abel-thm}. Let $\pi(x)$ be the number of primes less than or equal to~$x$. By the Prime Number Theorem, one knows that \[ \pi(n)\sim \frac{n}{\log n}.\] The Abelian Theorem permits us to conclude about the non-holonomic character of the sequence $(\pi(n))$, since \[ \sum_{n\ge1} \pi(n) z^n \mathop{\sim}_{z\to 1^-} \frac{1}{(1-z)^2 \log(1-z)^{-1}},\] which contradicts what the Structure Theorem permits. \begin{note} In this article, we concentrate on proofs of non-holonomicity based on analysis, that is, eventually, \emph{asymptotic approximations}. On a different register, powerful algebraic tools can be put to use in a number of situations. Considerations on power series have been used by Harris and Sibuya~\cite{HaSi85} to show: \emph{The reciprocal $(1/f)$ of a holonomic function~$f$ is holonomic if and only if~$f'/f$ is algebraic}. For instance, this proves the non-holonomicity of the reciprocal of Gauss' ${}_2F_1$ hypergeometric, except in degenerate cases. Using differential Galois theory, Singer generalized this result in~\cite{Singer86}. He characterized the possible polynomial relations between holonomic functions and also showed the following: \emph{A holonomic function~$f$ has to be algebraic if any of $\exp\int f$ or $\phi(f)$ is holonomic, with $\phi$ an algebraic function of genus~$\ge1$}. An analogous result for sequences is given in~\cite[Chap.~4]{SivdP97}: \emph{If both~$f_n$ and~$1/f_n$ are holonomic, then $f_n$ is an interlacing of hypergeometric sequences}. \end{note} \section{The logarithmic sequence} Let $f_n=\log n$ (with $\log 0\equiv0$) and let $f(z)$ be its generating function, \[ f(z)=\sum_{n\ge 1} (\log n) z^n.\] We propose to show that a sequence derived from $f_n$ by means of holonomicity preserving transformations is non-holonomic. Consider a variant of the $n$th difference of the sequence $f_n$, namely \[ \widehat f_n:=\sum_{k=1}^n \binom{n}{k} (-1)^k \log k,\] whose ordinary generating function $\widehat f(z):=\sum_{n\ge1} \widehat f_n z^n$ has positive radius of convergence and satisfies \begin{equation}\label{e2} \widehat f(z) = \frac{1}{1-z}f\left(-\frac{z}{1-z}\right). \end{equation} It is known that holonomic functions are closed under product and algebraic (hence also, rational) substitutions. Thus, $f$ and $\widehat f$ are such that either both of them are holonomic or none of them is holonomic. (See also Stanley's paper~\cite[p.~181]{Stanley80} for a discussion of the fact that differencing preserves holonomicity.) Next, Flajolet and Sedgewick proved in~\cite{FlSe95} that the sequence $\widehat f_n$ satisfies the asymptotic estimate \begin{equation}\label{e1} \widehat f_n = \log\log n+O(1). \end{equation} As a matter of fact, a full expansion is derived in~\cite[Th.~4]{FlSe95}, based on the N\"orlund-Rice integral representation~\cite{Norlund54} \begin{equation}\label{norlund} \widehat f_n = \frac{(-1)^n}{2i\pi}\int_{\mathcal H} (\log s) \frac{n!}{s(s-1)\cdots (s-n)}\, ds, \end{equation} the use of a Hankel contour~$\mathcal H$, and estimates akin to those used for the determination of inverse Mellin transforms affected with an algebraic-logarithmic singularity~\cite{Doetsch55}. % % % =\lfloor t^{-1}/\log t^{-1}\rfloor $. % The sum of the terms corresponding to $n