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A Family of Entire Functions Connecting the Bessel Function \(J_1\) and the Lambert W Function

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Abstract

Motivated by the problem of determining the values of \(\alpha >0\) for which \(f_\alpha (x)=\mathrm{e}^\alpha - (1+1/x)^{\alpha x},\ x>0\), is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family \(\varphi _\alpha \), \(\alpha >0\), of entire functions such that \(f_\alpha (x) =\int _0^\infty \mathrm{e}^{-sx}\varphi _\alpha (s)\,\mathrm{d}s, \ x>0.\) We show that each function \(\varphi _\alpha \) has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions \(\varphi _\alpha \), which turn out to be related to the well-known Bessel function \(J_1\) and the Lambert W function. On the other hand, by numerically evaluating the series expansion, we are able to show the behavior of \(\varphi _\alpha \) as \(\alpha \) increases from 0 to \(\infty \) and to obtain a very precise approximation of the largest \(\alpha >0\) such that \(\varphi _\alpha (s)\ge 0,\, s>0\), or equivalently, such that \(f_\alpha \) is completely monotonic.

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Acknowledgements

This work was initiated during a visit of the first author to the Department of Mathematics at the University of São Paulo in São Carlos, Brazil, in March 2018. He wants to thank the Department for generous support and hospitality during his stay. The second author was supported by: grant \(\#\)303447/2017-6, CNPq/Brazil. The third author was supported by: grant \(\#\)2016/09906-0, São Paulo Research Foundation (FAPESP). The authors thank a referee for useful references leading in particular to Remark 7.1.

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Appendix

Appendix

In this appendix we add a few remarks that came up after the submission of this work.

Remark 7.1

A referee has kindly pointed out that the coefficients \(c_{n,k}\) of the polynomials \(p_n(\alpha )\) can be expressed by the following formula:

$$\begin{aligned} c_{n,k}=(-1)^{n-k}\sum _{m=1}^k (-1)^m \frac{s(n+m,m)}{(n+m)!(k-m)!},\quad n\ge k\ge 1, \end{aligned}$$
(55)

where the s(pm) are the Stirling numbers of the first kind defined by

$$\begin{aligned} t(t-1)\cdots (t-p+1)=\sum _{m=0}^p s(p,m)t^m,\quad p\ge 1, \end{aligned}$$

\(s(0,0):=1\), see [6, p.278]. Note that \(s(p,0)=0\) for \(p\ge 1\), so in (55) one may sum from \(m=0\) as well. To see (55) we use the formula

$$\begin{aligned} B_n(a_1,\ldots ,a_n)=\sum _{k=1}^n B_{n,k}(a_1,\ldots ,a_{n-k+1}), \end{aligned}$$

where the partial Bell partition polynomials \(B_{n,k}\) are defined as

$$\begin{aligned} B_{n,k}(a_1,\ldots ,a_{n-k+1})=\sum _{J(n,k)}\frac{n!}{j_1!\cdots j_{n-k+1}!} \prod _{m=1}^{n-k+1}\left( \frac{a_m}{m!}\right) ^{j_m}, \end{aligned}$$

cf. [6, Section 11.2]. The sum is over the set J(nk) of all integers \(j_1,\ldots ,j_{n-k+1}\ge 0\) satisfying

$$\begin{aligned} j_1+\cdots +j_{n-k+1}=k,\quad j_1+2j_2+\cdots +(n-k+1)j_{n-k+1}=n. \end{aligned}$$

In the special case \(a_k=(-1)^k\alpha k!/(k+1),k=1,\ldots ,n\) we then get

$$\begin{aligned}&B_{n,k}\left( -\alpha \frac{1!}{2},\alpha \frac{2!}{3}, \ldots ,(-1)^{n-k+1}\alpha \frac{(n-k+1)!}{n-k+2}\right) \\&\quad =\sum _{J(n,k)}\frac{n!}{j_1!\cdots j_{n-k+1}!} \prod _{m=1}^{n-k+1}\left( \frac{(-1)^m\alpha }{m+1}\right) ^{j_m}\\&\quad =\alpha ^k(-1)^nB_{n,k}\left( \frac{1!}{2},\frac{2!}{3}, \ldots ,\frac{(n-k+1)!}{n-k+2}\right) . \end{aligned}$$

In [11, Theorem 1] one finds the evaluation

$$\begin{aligned} B_{n,k}\left( \frac{1!}{2},\frac{2!}{3},\ldots ,\frac{(n-k+1)!}{n-k+2}\right) =(-1)^{n-k}n!\sum _{m=1}^k\frac{(-1)^ms(n+m,m)}{(n+m)!(k-m)!}, \end{aligned}$$
(56)

and hence by (13),

$$\begin{aligned} p_n(\alpha )=\frac{(-1)^n}{n!}\sum _{k=1}^n \alpha ^k(-1)^n (-1)^{n-k}n!\sum _{m=1}^k\frac{(-1)^ms(n+m,m)}{(n+m)!(k-m)!}, \end{aligned}$$

and finally one obtains (55).

We observe that, from (55), it is possible to deduce the explicit formula for \(c_{n,1}\) given in Proposition 2.8-(i), using that \((-1)^ns(n+1,1)=n!\), and also, since \(s(n,2)=(-1)^n(n-1)! H_{n-1}\) with \(H_n=1+1/2+\cdots +1/n\) being the \(n^{th}\) harmonic number, to obtain the following formula for \(c_{n,2}\):

$$\begin{aligned} c_{n,2}=\frac{H_{n+1}}{n+2}-\frac{1}{n+1}. \end{aligned}$$

Further formulas can be obtained in terms of generalized harmonic numbers but they become increasingly more complicated.

Also the explicit formula for \(c_{n,n}\) given in Proposition 2.8-(i) can be obtained, by using (56) and the definition of \(B_{n,n}\):

$$\begin{aligned} c_{n,n}=\sum _{m=1}^n\frac{(-1)^m s(n+m,m)}{(n+m)!(n-m)!} =\frac{1}{n!}B_{n,n}\left( \frac{1}{2}\right) =\frac{1}{n!}\left( \frac{1}{2}\right) ^n. \end{aligned}$$

Remark 7.2

Alan Sokal asked the first author if Theorem 3.6 can be replaced by the stronger statement that \((p_n(\alpha ))_{n\ge 0}\) is a Hausdorff moment sequence when \(0\le \alpha \le 1\). The answer is yes, but the reader is warned that Eqs. (37) and (38) do not hold for \(n=-1\).

In fact, if \(0<\alpha <1\) we get for \(n=-1\),

$$\begin{aligned}&\frac{\mathrm{e}^{-\alpha }}{\pi }\int _0^1 (x/(1-x))^{\alpha x}\sin (\alpha \pi x) x^{-1}\,\mathrm{d}x =\mathrm{e}^{-\alpha }\lim _{x\rightarrow 0^+}f_\alpha (x)\\&\quad =\mathrm{e}^{-\alpha }(\mathrm{e}^{\alpha }-1)=1-\mathrm{e}^{-\alpha }<1=p_0(\alpha ), \end{aligned}$$

where we have used (29) and (11), and there is a similar calculation in case \(\alpha =1\). Using the Hausdorff moment sequence \((\delta _{n0})_{n\ge 0}=(1,0,0,0,\ldots )\) we find for \(0<\alpha <1\),

$$\begin{aligned} p_n(\alpha )=\mathrm{e}^{-\alpha }\delta _{n0}+\frac{\mathrm{e}^{-\alpha }}{\pi } \int _0^1 (x/(1-x))^{\alpha x}\frac{\sin (\alpha \pi x)}{x}x^n\,\mathrm{d}x,\quad n\ge 0, \end{aligned}$$

showing that \((p_n(\alpha ))_{n\ge 0}\) is a Hausdorff moment sequence when \(0<\alpha <1\). We similarly get \(p_n(0)=\delta _{n0}\) and

$$\begin{aligned} p_n(1)=\mathrm{e}^{-1}\delta _{n0}+\mathrm{e}^{-1}+\frac{\mathrm{e}^{-1}}{\pi }\int _0^1 (x/(1-x))^{x} \frac{\sin (\pi x)}{x}x^n\,\mathrm{d}x,\quad n\ge 0. \end{aligned}$$

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Berg, C., Massa, E. & Peron, A.P. A Family of Entire Functions Connecting the Bessel Function \(J_1\) and the Lambert W Function. Constr Approx 53, 121–154 (2021). https://doi.org/10.1007/s00365-020-09499-x

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