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A closed form for the Stirling polynomials in terms of the Stirling numbers. (English) Zbl 1432.11025

Summary: In the paper, by virtue of the Faà di Bruno formula and two identities for the Bell polynomial of the second kind, the authors find a closed form for the Stirling polynomials in terms of the Stirling numbers of the first and second kinds.

MSC:

11B73 Bell and Stirling numbers
11B68 Bernoulli and Euler numbers and polynomials
33B10 Exponential and trigonometric functions

Software:

Stirling
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References:

[1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. · Zbl 0283.05001
[2] B.-N. Guo, I. Mező, and F. Qi, An explicit formula for the Bernoulli polynomials in terms of the r-Stirling numbers of the second kind, Rocky Mountain J. Math. 46 (2016), no. 6, 1919-1923; Available online at http://dx.doi.org/10.1216/RMJ-2016-46-6-1919. · Zbl 1371.11045
[3] B.-N. Guo and F. Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. Anal. Number Theory 3 (2015), no. 1, 27-30; Available online at http://dx.doi.org/10.12785/jant/030105.
[4] B.-N. Guo and F. Qi, On inequalities for the exponential and logarithmic functions and means, Malays. J. Math. Sci. 10 (2016), no. 1, 23-34.
[5] B.-N. Guo and F. Qi, Some identities and an explicit formula for Bernoulli and Stirling num- bers, J. Comput. Appl. Math. 255 (2014), 568-579; Available online at http://dx.doi.org/10.1016/j.cam.2013.06.020. · Zbl 1291.11051
[6] C. Jordan, Calculus of Finite Differences, Chelsea, New York, 1965. · Zbl 0154.33901
[7] N. Nielsen, Gammafunktionen, Leipzig, 1906.
[8] N. E. Nórlund, Vorlesungen ’uber Differenzenrechnung, Springer-Verlag, Berlin, 1924.
[9] F. Qi, A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, Publ. Inst. Math. (Beograd) (N.S.) 100 (114) (2016), 243-249; Available online at http://dx.doi.org/10.2298/PIM150501028Q. · Zbl 1432.11023
[10] F. Qi, Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind, Math. Inequal. Appl. 19 (2016), no. 1, 313-323; Available online at http://dx.doi.org/10.7153/mia-19-23. · Zbl 1333.11027
[11] F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat 28 (2014), no. 2, 319-327; Available online at http://dx.doi.org/10.2298/FIL1402319O. · Zbl 1385.11011
[12] F. Qi and R. J. Chapman, Two closed forms for the Bernoulli polynomials, J. Number Theory 159 (2016), 89-100; Available online at http://dx.doi.org/10.1016/j.jnt.2015.07.021. · Zbl 1400.11070
[13] F. Qi and B.-N. Guo, A closed form for the Stirling polynomials in terms of the Stirling numbers, Preprints 2017, 2017030055, 4 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0055.v1. · Zbl 1432.11025
[14] F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Article 140, 14 pages; Available online at http://dx.doi.org/10.1007/s00009-017-0939-1. · Zbl 1427.11023
[15] A. Schreiber, Multivariate Stirling polynomials of the first and second kind, Discrete Math. 338 (2015), no. 12, 2462-2484; Available online at http://dx.doi.org/10.1016/j.disc.2015. 06.008. · Zbl 1321.11030
[16] M. Ward, The representation of Stirling’s numbers and Stirling’s polynomials as sums of factorials, Amer. J. Math. 56 (1934), no. 1-4, 87-95; Available online at http://dx.doi.org/10.2307/2370916. · JFM 60.0300.01
[17] Z.-Z. Zhang and J.-Z. Yang, Notes on some identities related to the partial Bell polynomials, Tamsui Oxf. J. Inf. Math. Sci. 28 (2012), no. 1, 39-48.
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