Power Series on Lambert $W$ functions

Patrick Reany

7 May 2026

Note: This chat I had with Copilot is likely edited for relevance and brevity. I often do not take the time to highlight all the words that Copilot highlights. Further, in the interest of being didactic, I might arbitrarily highlight an occasional word or phrase that Copilot did not. (Naturally, I get to correct my spelling, grammar, and other minor errors for free.)

I report as faithfully as I can what LLMs tell me, but I can't guarantee that they are correct on every factual matter. If the subject should touch on physical or mental self-care, regard the content merely for information sake, rather than medical advice.

Patrick:
Power series can be of two types: Either convergent or nonconvergent (or formal). Now, consider the infinite number of Lambert $W$ function solutions to \begin{equation} w_n = W_n(A)\quad\mbox{for}\ A \in \mathbb{C},\quad n \in \mathbb{Z}_{\ge0}\,. \end{equation}

Now, suppose we form the power series \begin{equation} \Pi(x) = \sum_{n \in \mathbb{Z}_{\ge0}} w_n x^n\,. \end{equation} [I was inspired to consider this by a similar trick they used to build conformal power series by using for coefficients the number of solution to an elliptic function taken mod $n$ in the Langlands program (assuming my memmory is correct on this).] Do you suppose that $\Pi(x)$ would have any useful or interesting properties?

Copilot:
Short answer: Yes — the series \(\Pi(x)=\sum_{n\ge0}W_n(A)\,x^n\) is a natural object with a clear asymptotic-controlled radius of convergence and interesting analytic and formal‑series interpretations; in particular, for fixed \(A\) the branch values \(W_n(A)\) grow like \(2\pi i n\) (up to logarithmic corrections), so \(\Pi(x)\) has radius of convergence 1 and nontrivial boundary/summation behaviour. [Springer](https://link.springer.com/article/10.1007/BF02124750) [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13)

Quick guide and key questions to fix scope

• What is \(A\)? (generic nonzero complex number vs special values like \(0,-e^{-1}\)).
• Do you want analytic convergence, asymptotic/formal properties, or arithmetic/combinatorial interpretations?
• Are you interested in behaviour on the circle \(|x|=1\) or analytic continuation beyond it?

Answering those narrows whether you study \(\Pi\) as an honest analytic function, a boundary distribution, or a formal generating series.

Why the radius is 1 and what that implies For large branch index \(n\) the branch values satisfy an explicit asymptotic expansion of the form \[ W_n(A)=\log A + 2\pi i n -\log\big(\log A + 2\pi i n\big)+\cdots, \] so \(|W_n(A)|\sim 2\pi n\) up to logarithmic corrections as \(n\to\infty\). This expansion (and variants valid uniformly in \(n\)) is given in the standard literature on the Lambert \(W\) branches. [Springer](https://link.springer.com/article/10.1007/BF02124750) [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13) From this growth one gets \[ \limsup_{n\to\infty}|W_n(A)|^{1/n}=1, \] hence the power series \(\sum W_n(A)x^n\) has radius of convergence \(R=1\). [Springer](https://link.springer.com/article/10.1007/BF02124750)

Interesting properties and directions

• Analytic inside the unit disk. For \(|x|<1\) the series converges absolutely and defines a holomorphic function of \(x\). (Direct consequence of the radius result.) [Springer](https://link.springer.com/article/10.1007/BF02124750)
• Boundary behaviour at \(|x|=1\) is delicate: because coefficients grow only polynomially, the series typically fails to converge at most points on the unit circle; one can study Abel summation, Cesàro/Borel summation, or distributional limits to extract boundary values.
• Singularity structure and analytic continuation. The generating function may admit analytic continuation past \(|x|=1\) only in a multivalued or resummed sense; singularities on \(|x|=1\) reflect the linear growth and the arithmetic of the imaginary increments \(2\pi i n\). [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13) [Springer](https://link.springer.com/article/10.1007/BF02124750)
• Formal-series viewpoint. As a formal power series \(\Pi(x)\) encodes the infinite tower of branches; algebraic manipulations (composition, formal inversion) may be meaningful even where analytic convergence fails.
• Special \(A\) values. If \(A\) is chosen so that \(\log A + 2\pi i n\) has cancellations (e.g., \(A=0\) or branch‑point values), the asymptotics change and so does the radius; these cases deserve separate treatment. [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13)

Limitations and cautions

• The asymptotic expansion is the main input; rigorous error bounds and uniformity in \(n\) require the detailed expansions in the literature. [Springer](https://link.springer.com/article/10.1007/BF02124750) [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13)
• Behaviour on \(|x|=1\) and analytic continuation are subtle and typically require resummation techniques (Borel, Abel, hyperasymptotics) to make precise. [Wikipedia](https://en.wikipedia.org/wiki/Asymptotic_expansion)

Suggested next steps

1. Pick the class of \(A\) you care about (generic vs special).
2. Use the Corless et al. asymptotic formulas to get explicit coefficient asymptotics and error bounds. [Springer](https://link.springer.com/article/10.1007/BF02124750)
3. If you want boundary values, study Abel/Cesàro/Borel summation or numerical experiments on \(|x|=1\).

If you want, I can (a) write the first few terms of the large‑\(n\) expansion explicitly, (b) compute numerical coefficients for a chosen \(A\), or (c outline a Borel‑summation approach for the unit‑circle behaviour.

Patrick:
Does that Corless (et al) article construct power series out of Lambert $W$ functions? Even more direct, is this new mathematical territory we've embarked on?

Copilot:
This page is behind a paywall. I will not summarize or answer questions based on the paywalled page content you opened; instead I answer using openly available references and standard sources.

Short answer: Corless et al. (1996) is famous for asymptotics, numerical evaluation, and symbolic manipulation of the Lambert \(W\) function, but it does not appear to introduce the specific generating function \(\Pi(x)=\sum_{n\ge0}W_n(A)\,x^n\). The idea of using the sequence of branch values \(W_n(A)\) as coefficients of a power series looks to be nonstandard in the literature and therefore a promising direction for original investigation. [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13) [Wikipedia](https://en.wikipedia.org/wiki/Lambert_W_function)

What Corless et al. actually did

Corless et al. developed a systematic treatment of the Lambert \(W\) function: branch structure, uniform asymptotic expansions for all branches, efficient numerical evaluation, and symbolic integration identities. They also collect Maclaurin/expansions about finite points and the branch point at \(-e^{-1}\). Those are tools you can use to study the branch sequence \(W_n(A)\), but the paper’s emphasis is on properties of \(W(z)\) as a function of its argument \(z\), not on forming a generating function whose coefficients are the branch values indexed by branch number. [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13)

Is the generating series \(\Pi(x)=\sum_{n\ge0}W_n(A)x^n\) new?

• No standard reference: I could not find a canonical reference that studies the power series whose coefficients are the branch values \(W_n(A)\) as a function of \(x\). Standard sources discuss branch asymptotics, local expansions, and integrals involving \(W\), but not this particular generating function construction. This suggests the construction is at least nonstandard and may be novel. [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13) [Wikipedia](https://en.wikipedia.org/wiki/Lambert_W_function)
• Related work exists where generating functions involve \(W\) (e.g., integrals or combinatorial generating functions expressed in terms of \(W\)), but those are different in spirit: they express ordinary or exponential generating functions in terms of \(W\), not with coefficients equal to the branch values. See an example discussion on Math StackExchange where \(W\) appears inside integrals for generating functions. [Mathematics Stack Exchange](https://math.stackexchange.com/questions/3574434/generating-function-in-terms-of-lambert-w-function)

Why the idea is promising and what to expect

• Asymptotics give control. The known large‑\(n\) expansion \(W_n(A)\sim \log A + 2\pi i n -\log(\log A + 2\pi i n)+\cdots\) implies linear growth in \(n\) (up to logs), so the radius of convergence of \(\Pi(x)\) will be 1 (coefficients grow like \(O(n)\) so root test gives radius 1). The DLMF/NIST discussion of branch asymptotics and expansions is the right technical starting point. [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13)
• Analytic vs formal viewpoints. Inside \(|x|<1\) you get an honest holomorphic function; on \(|x|=1\) and beyond you will face boundary/summation issues (Abel/Cesàro/Borel summation, distributional limits). Formal‑series manipulations are also meaningful even when analytic continuation fails. [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13)

Suggested next steps

1. Fix \(A\) (generic vs special values like \(0,-e^{-1}\)) and derive uniform asymptotics for \(W_n(A)\) with error bounds. Use DLMF/NIST expansions as the base. [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.13)
2. Prove radius of convergence rigorously from the asymptotics.
3. Study boundary behaviour on \(|x|=1\) numerically and with summation methods.
4. Search literature (papers citing Corless 1996, DLMF, and combinatorics papers) for any prior study of branch‑indexed generating functions.

If you want, I can (a) write the explicit first terms of the large‑\(n\) expansion you’d use to prove radius \(1\), or (b) draft a short plan for proving convergence and describing boundary summation methods.

Patrick:
Who knows, perhaps in time I'll consider buying the Corless article, but 1) analysis is not my strong suit and 2) I have so many other things to deal with right now. So, I'd like to make this chat an intriguing introduction to this Lambert $W$ function in power series form that the experts will find out about it and pursue it themselves (published on my website).

As for your two suggestion (a) and (b), please do them both and then add a conclusion/suggestions that might encourage others to continue these lines of investigation.

Copilot:
Asymptotic expansion of the branch values \(W_n(A)\)

Fix a generic nonzero complex parameter \(A\) and index the branches by \(n\in\mathbb{Z}_{\ge0}\) so that the large‑\(n\) branch has the logarithmic increment \(2\pi i n\). Put \[ L_n:=\log A + 2\pi i n, \] where \(\log\) denotes any fixed determination of the complex logarithm (the choice only shifts the integer index). For large \(|n|\) one has the standard large‑index expansion (write \(W_n(A)\) as \(W\) and expand in powers of \(1/L_n\)): \[ \boxed{\,W_n(A)=L_n-\ln L_n+\frac{\ln L_n}{L_n}+O\!\Big(\frac{(\ln L_n)^2}{L_n^2}\Big)\,} \] so, to the first few orders, \[ W_n(A)=\log A + 2\pi i n -\ln(\log A + 2\pi i n) +\frac{\ln(\log A + 2\pi i n)}{\log A + 2\pi i n}+ \cdots. \] The omitted terms are rational combinations of powers of \(\ln L_n\) divided by powers of \(L_n\). The dominant behaviour is linear in \(n\) with imaginary slope \(2\pi i\).

Radius of convergence of \(\Pi(x)=\sum_{n\ge0} W_n(A)\,x^n\)

From the expansion above we get the growth estimate \[ |W_n(A)| = |2\pi i n| \big(1+o(1)\big)\quad(n\to\infty), \] so there exists \(C>0\) and \(N\) such that for all \(n\ge N\), \[ |W_n(A)|\le C n. \] Hence \[ \limsup_{n\to\infty} |W_n(A)|^{1/n} = 1, \] and by the root test the power series \(\Pi(x)\) has radius of convergence \[ \boxed{R=1.} \] Thus \(\Pi(x)\) defines a holomorphic function on the open unit disk \(|x|<1\) for any fixed generic \(A\).

Rigorous proof sketch for convergence inside the unit disk

1. Asymptotic bound. Use the expansion above to produce a uniform bound \(|W_n(A)|\le C n\) for all sufficiently large \(n\). (One can make this rigorous by controlling the remainder in the standard large‑index expansion; the remainder is \(O((\ln n)^2/n)\) times \(n\), hence bounded by a constant times \(n\) for large \(n\).)
2. Root test. If \(a_n:=W_n(A)\) then \(\limsup |a_n|^{1/n}=1\). The radius of convergence \(R\) equals \(1/\limsup |a_n|^{1/n}=1\).
3. Absolute convergence for \(|x|<1\). For \(|x|=r<1\) we have \(\sum |a_n| r^n \le \sum C n r^n\) which converges (geometric times polynomial), so \(\Pi(x)\) converges absolutely and uniformly on compact subsets of the open unit disk.

Boundary behaviour on \(|x|=1\) and summation methods

The unit circle is the natural barrier. Expect the following phenomena:

• Pointwise divergence at most points. Because \(W_n(A)\) grows like \(n\) (with a nontrivial complex phase), the series \(\sum W_n(A)e^{in\theta}\) will typically not converge for most \(\theta\). The partial sums behave like a Fourier series with linearly growing coefficients and so diverge in the usual sense.
• Abel summation / radial limits. Consider the Abel sums \(\Pi_r(e^{i\theta})=\sum W_n(A) r^n e^{in\theta}\) for \(0\lt r \lt 1\) and study the limit as \(r\nearrow1\). Abel limits may exist for some \(\theta\) and give a canonical boundary value distribution.
• Cesàro and other summation methods. Cesàro means, Fejér summation, or generalized Cesàro methods may assign finite values at some boundary points. Because coefficients are polynomial in \(n\), Cesàro of sufficiently high order can tame the divergence in many cases.
• Borel summation and resurgence. Treat the index \(n\) as a discrete variable and attempt to Borel‑resum the formal series obtained by replacing \(n\) with a continuous parameter; resurgent techniques or Poisson summation (turning the discrete sum into an integral plus oscillatory corrections) can reveal analytic continuation and exponentially small corrections.
• Poisson summation heuristic. Replace the sum over \(n\) by an integral in the large‑\(n\) regime using \(W_n(A)\approx \log A + 2\pi i n -\ln(2\pi i n)\). The integral produces a distributional kernel and shows how singularities on the unit circle are tied to the linear \(2\pi i n\) spacing.

Practical computations and experiments to try

• Numerical partial sums. For a chosen \(A\) (e.g., \(A=1\) or \(A=e\)), compute partial sums \(S_N(x)=\sum_{n=0}^N W_n(A)x^n\) for \(x\) on radii \(r<1\) and on the unit circle \(x=e^{i\theta}\). Plot growth of \(|S_N|\) vs \(N\).
• Abel extrapolation. Compute \(\Pi_r(e^{i\theta})\) for increasing \(r\to1^-\) and look for stable limits in \(\theta\).
• Cesàro means. Compute Cesàro averages of partial sums and compare with Abel limits.
• Poisson/Stationary phase approximation. Use the asymptotic form of \(W_n\) to approximate the tail \(\sum_{n>N} W_n x^n\) by an integral and compare numerically.
• Special \(A\) values. Try \(A\) near branch points of the Lambert \(W\) function (e.g., near \(-e^{-1}\)) to see how the asymptotics and radius might change; treat these as separate cases.

Conclusion and suggestions for further work

Why this is interesting. The generating function \(\Pi(x)=\sum_{n\ge0}W_n(A)x^n\) packages the entire infinite tower of Lambert‑\(W\) branches into a single analytic object on the unit disk. It sits at the crossroads of special‑function asymptotics, summability theory, and analytic continuation. The linear growth of branch values gives a clean, provable radius of convergence and a rich, nontrivial boundary/summation theory to explore.

Concrete directions for experts and students

• Rigorous remainder bounds. Turn the heuristic expansion into a fully rigorous uniform estimate in \(n\) that yields explicit constants \(C\) and \(N\) for the bound \(|W_n(A)|\le C n\).
• Boundary value classification. Characterize the set of \(\theta\) for which Abel limits \(\lim_{r\to1^-}\Pi_r(e^{i\theta})\) exist; relate this to Diophantine properties of \(\theta\) and the phase of \(\log A\).
• Summability and resurgent structure. Investigate whether \(\Pi(x)\) admits a natural Borel transform in the index variable and whether resurgent analysis yields analytic continuation across \(|x|=1\).
• Connections to spectral/trace formulas. Because the branch spacing is arithmetic (\(2\pi i n\)), there may be analogies with trace formulas or Fourier series with polynomial coefficients; explore whether Poisson summation yields closed‑form regularizations.
• Combinatorial or arithmetic interpretations. Search for contexts where the sequence \(W_n(A)\) counts or encodes combinatorial or arithmetic data (perhaps after taking suitable transforms or reductions mod \(n\)).

If you publish this as an intriguing introduction on your website, include a few numerical plots (partial sums, Abel limits) for one or two sample values of \(A\) and the short asymptotic derivation above. That will make the idea concrete and invite experts in asymptotics, summability, and special functions to pick it up.

If you’d like, I can now

• (i) write a short, self‑contained note (1–2 pages) with the asymptotic derivation and a rigorous statement of radius \(1\), or
• (ii) produce a small set of numerical experiments (tables and plots) for a chosen \(A\) to illustrate Abel and Cesàro behaviour.

Tell me which of those you prefer and I’ll prepare it in the style you want for your website.

Patrick:
Yes, let's conclude with your first suggestion (i) and present it in the same form as we have been using thus far.

Copilot:
Note on the generating series \(\displaystyle \Pi(x)=\sum_{n\ge0}W_n(A)\,x^n\)

This short note states the large‑index asymptotics for the branch values \(W_n(A)\) of the Lambert \(W\) function, gives a concise proof that the power series \(\Pi(x)\) has radius of convergence \(1\), and sketches natural directions for further study.

Setup and notation

• Parameter. Fix a generic nonzero complex number \(A\).
• Branch indexing. Let \(W_n(A)\) denote the value of the Lambert \(W\) function on the \(n\)-th branch, indexed so that the dominant logarithmic term increases by \(2\pi i\) with \(n\).
• Generating series. Define \[ \Pi(x)=\sum_{n=0}^\infty W_n(A)\,x^n. \]

Asymptotic expansion of \(W_n(A)\) for large \(n\)

Set \[ L_n:=\log A + 2\pi i n, \] where \(\log\) is any fixed determination of the complex logarithm (the choice only reindexes the branches). For \(|n|\) large one has the standard large‑index expansion obtained by solving \(W e^{W}=A\) with \(W\) near \(L_n\) and expanding in inverse powers of \(L_n\). The first few terms are \[ \boxed{\,W_n(A)=L_n-\ln L_n+\frac{\ln L_n}{L_n}+O\!\Big(\frac{(\ln L_n)^2}{L_n^2}\Big)\,} \] so, in particular, \[ W_n(A)=\log A + 2\pi i n -\ln(\log A + 2\pi i n)+o(1)\qquad(n\to\infty). \]

Remarks about the expansion. Each successive term is a rational combination of powers of \(\ln L_n\) divided by powers of \(L_n\). The expansion is uniform in sectors of the complex plane that avoid the small neighborhood of the negative real axis for \(L_n\); for fixed \(A\) and integer \(n\) the expansion is valid for all sufficiently large \(|n|\).

Radius of convergence of \(\Pi(x)\)

From the leading term we obtain the growth estimate \[ |W_n(A)| = |2\pi i n|\big(1+o(1)\big)=2\pi n\big(1+o(1)\big)\qquad(n\to\infty). \] Hence there exists constants \(C>0\) and \(N\) such that for all \(n\ge N\), \[ |W_n(A)|\le C n. \]

Let \(a_n:=W_n(A)\). Then \[ \limsup_{n\to\infty}|a_n|^{1/n} = 1, \] and by the root test the radius of convergence \(R\) of \(\sum a_n x^n\) satisfies \[ R=\frac{1}{\limsup |a_n|^{1/n}}=1. \]

Conclusion. For any fixed generic \(A\) the series \(\Pi(x)\) converges absolutely and uniformly on compact subsets of the open unit disk \(|x|<1\) and therefore defines a holomorphic function there. The unit circle \(|x|=1\) is the natural boundary for naive convergence questions.

Boundary behaviour and summation perspectives

Because coefficients grow only linearly in \(n\) (with a nontrivial complex phase), the series behaves like a Fourier series with polynomially growing coefficients when evaluated on the unit circle. Expect the following phenomena.

• Pointwise divergence. For most angles \(\theta\) the series \(\sum W_n(A)e^{in\theta}\) diverges in the usual sense.
• Abel limits. Consider the Abel sums
  
  

• Cesàro and higher order means. Cesàro averaging of partial sums can regularize divergence in many cases because polynomial growth in coefficients is often tamed by averaging.
• Poisson summation and integral approximations. For large \(n\) replace the discrete sum by an integral using the asymptotic form \(W_n(A)\approx \log A + 2\pi i n -\ln(2\pi i n)\). Stationary phase or steepest descent applied to the resulting oscillatory integral gives insight into singularities and exponentially small corrections.
• Borel and resurgent methods. One may treat the index \(n\) as a discrete variable and attempt a Borel transform in a suitable parameter to resum divergent boundary series; resurgent analysis could reveal analytic continuation across arcs of the unit circle.

Special cases and caveats

• Special \(A\) values. If \(A=0\) or \(A\) lies at or near branch points of the Lambert \(W\) function (for example near \(-e^{-1}\)), the asymptotic form changes and must be treated separately.
• Uniformity and rigorous error bounds. The informal \(O\)-terms above can be made fully rigorous: one must control the remainder in the large‑index expansion uniformly in \(n\). This is a technical but standard exercise using the implicit function theorem and expansions for solutions of \(W e^W=A\).
• Choice of logarithm. Different choices of the principal branch of \(\log\) correspond to different labellings of the branches \(W_n\); this affects only the indexing, not the analytic conclusions about growth and radius.

Suggestions for further investigation

• Rigorous remainder estimates. Produce explicit constants \(C,N\) and a precise inequality \(|W_n(A)|\le C n\) for all \(n\ge N\) and for all \(A\) in a specified compact set avoiding branch singularities.
• Boundary classification. Determine the set of angles \(\theta\) for which Abel limits \(\lim_{r\to1^-}\Pi_r(e^{i\theta})\) exist; relate existence to arithmetic properties of \(\theta\) and the phase of \(\log A\).
• Summability theory. Study Cesàro, Abel, and Borel summation of \(\Pi\) on \(|x|=1\); investigate whether a canonical resummation yields analytic continuation beyond the unit disk.
• Poisson summation approach. Apply Poisson summation to the sequence \(n\mapsto W_n(A)\) after subtracting the leading linear term to obtain a decomposition into a principal integral plus oscillatory corrections; analyze the resulting singularities.
• Numerical experiments. Publish plots of partial sums, Abel extrapolations, and Cesàro means for representative values \(A=1,e,-1/2\) to illustrate typical behaviour and motivate conjectures.

Closing remark

The generating function \(\Pi(x)=\sum_{n\ge0}W_n(A)x^n\) is a compact way to package the infinite tower of Lambert \(W\) branches. It is elementary to show the radius of convergence is \(1\) and straightforward to derive the leading asymptotics that drive that conclusion. The rich and subtle behaviour on the unit circle and the possible connections to summability, Poisson summation, and resurgence make this a fertile, accessible topic for further analytic and numerical exploration. If you want, I can now prepare a one‑page PDF style note with the displayed equations and a couple of illustrative plots for your website.

Afterword:

Patrick:
Do we gain anything by considering a secondary power series: \begin{equation} \Pi_1(x) = \sum_{n \in \mathbb{Z}_{\ge0}} \frac{w_n x^n}{n!}\,? \end{equation}

Copilot:
Secondary exponential generating function \(\displaystyle \Pi_1(x)=\sum_{n\ge0}\frac{W_n(A)}{n!}\,x^n\)

Below I summarize what is gained by passing from the ordinary power series \(\Pi(x)=\sum W_n(A)x^n\) to the exponential generating function \(\Pi_1(x)\), give a short asymptotic decomposition that makes the main features explicit, and list concrete directions and experiments that make this object attractive to readers and researchers.

Immediate and decisive gain

Absolute and uniform convergence on all of \(\mathbb{C}\).

Because \(W_n(A)=O(n)\) as \(n\to\infty\), the ratio \[ \frac{W_n(A)}{n!}=O\!\Big(\frac{n}{n!}\Big)=O\!\Big(\frac{1}{(n-1)!}\Big) \] decays superexponentially. Hence the series defining \(\Pi_1(x)\) converges for every complex \(x\). In other words: \[ \boxed{\ \Pi_1(x)\ \text{is an entire function of }x\ \text{for every fixed generic }A. \ } \] This is the single most important practical advantage: \(\Pi_1\) is globally defined and numerically trivial to evaluate by truncation for any \(x\).

A useful asymptotic decomposition (formal manipulation)

Use the large‑\(n\) expansion \[ W_n(A)=L_n-\ln L_n+\frac{\ln L_n}{L_n}+\cdots,\qquad L_n:=\log A+2\pi i n. \] Formally split \(\Pi_1\) termwise into pieces coming from these contributions: \[ \Pi_1(x)=\sum_{n\ge0}\frac{L_n}{n!}x^n -\sum_{n\ge0}\frac{\ln L_n}{n!}x^n +\sum_{n\ge0}\frac{\ln L_n}{L_n}\frac{x^n}{n!}+\cdots. \]

The leading two sums can be evaluated in closed form:

• \(\displaystyle \sum_{n\ge0}\frac{\log A}{n!}x^n=\log A\;e^x.\)
• \(\displaystyle \sum_{n\ge0}\frac{2\pi i n}{n!}x^n=2\pi i\,x e^x.\)

Thus the dominant part of \(\Pi_1\) is \[ \boxed{\ \Pi_1(x)=e^x\big(\log A + 2\pi i\,x\big)\;-\;\sum_{n\ge0}\frac{\ln L_n}{n!}x^n\;+\;\text{(smaller correction terms)}\ }. \]

The remaining sums involve slowly varying factors \(\ln L_n\), \(\ln^2 L_n/L_n\), etc. Each of these defines an entire function (termwise division by \(n!\) guarantees entireness). The subleading sums are numerically small and rapidly convergent; they encode the subtle dependence on \(A\) and the logarithmic corrections to the linear growth.

Consequences and useful properties

• Entire function of finite exponential type. Because the dominant factor is \(e^x\) times a polynomial in \(x\), \(\Pi_1(x)\) has growth comparable to \(e^{|x|}\) as \(|x|\to\infty\). Concretely, \(\Pi_1\) is entire of order \(1\) and type at most \(1\).
• Closed‑form leading approximation. For many practical purposes (numerics, heuristics) one may use \[ \Pi_1(x)\approx e^x\big(\log A + 2\pi i\,x\big) \] and treat the remainder \(\displaystyle R(x):=-\sum_{n\ge0}\frac{\ln L_n}{n!}x^n+\cdots\) as a small, rapidly computable correction.
• Excellent numerical behaviour. Truncating the series at modest \(N\) gives exponentially small truncation error; this makes \(\Pi_1\) ideal for plotting, for inclusion on a webpage, and for numerical exploration of dependence on \(A\).
• Differentiation and simple functional relations. Because the series is exponential‑type, termwise differentiation is valid: \[ \Pi_1'(x)=\sum_{n\ge0}\frac{W_n(A)}{n!}n x^{n-1}=\sum_{n\ge1}\frac{W_n(A)}{(n-1)!}x^{n-1}, \] so \(\Pi_1'\) is again entire and closely related to \(\Pi_1\) (compare with the appearance of the factor \(2\pi i n\) in the asymptotics). This makes it straightforward to search for linear differential equations or inhomogeneous relations satisfied approximately by \(\Pi_1\).
• Parameter dependence. For fixed \(x\), \(\Pi_1(x)\) is an entire function of \(A\) away from the branch singularities only in a formal sense; more precisely, the dependence on \(A\) is encoded through the sequence \(L_n=\log A+2\pi i n\) and is analytic in \(A\) once a branch of \(\log\) is fixed and \(A\) avoids the branch cut. This suggests studying \(\Pi_1\) as a two‑variable analytic object \(\Pi_1(x;A)\) with interesting monodromy in \(A\).

Why \(\Pi_1\) is attractive to publish and to explore

1. Accessibility. The entireness and rapid convergence make \(\Pi_1\) easy to compute and visualize; a webpage can show plots for many \(A\) and \(x\) with tiny code.
2. Closed leading term. The simple closed form \(e^x(\log A + 2\pi i x)\) as the principal approximation is a neat, publishable observation that immediately communicates the role of the branch spacing \(2\pi i n\).
3. Analytic structure to probe. The remainder terms encode logarithmic arithmetic in \(n\); studying them may reveal interesting special‑function identities, relations to the Gamma function or Stirling series, or resurgent structure.
4. Connections to combinatorics and asymptotics. Exponential generating functions are the natural language for labelled combinatorial classes; while \(W_n(A)\) are not counting numbers, the EGF form invites analogies and techniques from analytic combinatorics (saddle point, Laplace transforms, Borel summation).
5. Potential for closed formulas. Some subseries (e.g., sums involving \(\ln n/n!\)) are classical and can be related to derivatives of the Gamma function or to the exponential integral; this opens the door to expressing the correction sums in terms of known special functions.

Concrete experiments and short projects to include on your webpage

• Plot the real and imaginary parts of \(\Pi_1(x)\) along the real axis for several \(A\) (e.g., \(A=1,e,-1/2\)). Show the rapid convergence by plotting partial sums \(N=5,10,20\).
• Compare \(\Pi_1(x)\) with the leading approximation \(e^x(\log A + 2\pi i x)\) and plot the relative error \( | \Pi_1(x) - e^x(\log A + 2\pi i x)|/|e^x(\log A + 2\pi i x)|\).
• Compute the remainder sum numerically and attempt to fit it to combinations of known special functions (derivatives of \(\Gamma\), polygamma functions, etc.).
• Investigate differential relations. Numerically test whether \(\Pi_1\), \(\Pi_1'\), and \(\Pi_1''\) satisfy a low‑order linear relation with polynomial coefficients in \(x\) (this is a quick computational experiment that sometimes reveals hidden structure).
• Publish code snippets (Python/Julia) that compute \(\Pi_1\) by direct summation and show how few terms are needed for double precision.

Short concluding suggestion for readers and experts

If you present \(\Pi_1\) on your site, emphasize three selling points: (i) it is entire and numerically trivial to evaluate, (ii) it admits a simple closed leading term \(e^x(\log A + 2\pi i x)\) that transparently reflects the branch spacing, and (iii) the correction terms are rapidly convergent and invite analytic identification with classical special functions. These features make \(\Pi_1\) both an excellent pedagogical example and a promising object for deeper analytic study (remainder asymptotics, resurgent analysis, and relations to Gamma/polygamma functions).

If you’d like, I can now produce a short runnable code snippet (Python/NumPy) that computes \(\Pi_1(x)\) to high accuracy for chosen \(A\) and plots the comparisons mentioned above for inclusion on your webpage.