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.
Similar content being viewed by others
References
Alzer, H., Berg, C.: Some classes of completely monotonic functions. Ann. Acad. Sci. Fenn. Math. 27, 445–460 (2002)
Berg, C.: Problem 1. Bernstein functions. J. Comput. Appl. Math. 178, 525–526 (2005)
Berg, C.: Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity. In: Mateu, J., Porcu, E. (eds.) Positive Definite Functions: From Schoenberg to Space-Time Challenges, pp. 15–45. Citeseer, Castellón de la Plana (2008)
Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 87. Springer, Berlin (1975)
Boas, R.P.: Entire Functions. Academic Press, New York (1954)
Charalambides, C.A.: Enumerative Combinatorics. Chapman & Hall/CRC, Boca Raton (2002)
Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert \(W\) function. Adv. Comput. Math. 5, 329–359 (1996)
Guo, B.-N., Qi, F.: A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function. U.P.B. Sci. Bull. Ser. A 72(2), 21–30 (2010)
Qi, F., Li, W., Guo, B.-N.: Generalizations of a theorem of Schur. Appl. Math. E-Notes 6, 244–250 (2006). Art. 29
Qi, F., Niu, D.-W., Cao, J.: Logarithmically completely monotonic functions involving gamma and polygamma functions. J. Math. Anal. Approx. Theory 1(1), 66–74 (2006)
Qi, F.: Diagonal recurrence relations for the Stirling numbers of the first kind. Contrib. Discrete Math. 11(1), 22–30 (2016)
Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill Book Co, London (1976)
Schilling, R.L., Song, R., Vondraček, Z.L: Bernstein Functions: Theory and Applications. De Gruyter Studies in Mathematics, vol. 37. de Gruyter, Berlin (2010)
Shemyakova, E., Khashin, S.I., Jeffrey, D.J.: A conjecture concerning a completely monotonic function. Comput. Math. Appl. 60, 1360–1363 (2010)
Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mourad Ismail.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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:
where the s(p, m) are the Stirling numbers of the first kind defined by
\(s(0,0):=1\), 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
where the partial Bell partition polynomials \(B_{n,k}\) are defined as
cf. [6, Section 11.2]. The sum is over the set J(n, k) of all integers \(j_1,\ldots ,j_{n-k+1}\ge 0\) satisfying
In the special case \(a_k=(-1)^k\alpha k!/(k+1),k=1,\ldots ,n\) we then get
In [11, Theorem 1] one finds the evaluation
and hence by (13),
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 \(c_{n,2}\):
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 \(B_{n,n}\):
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 \(n=-1\),
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 \(0<\alpha <1\),
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
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-020-09499-x
Keywords
- Completely monotonic function
- Complex analysis
- Fourier analysis
- Stieltjes moment sequence
- Bell polynomials