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They were called exponential polynomials also by J-Carlo Rota and other researches. Under the name “exponential polynomials” they appeared in a number of publications, including my papers (see below). Most properties were found by
Grunert, Bell, and in the papers listed below, especially in , , , and . Most of these papers are available online. Touchard has not much contribution to their theory and was not the first one to mention them. He was just one of the many authors. It is not clear at all how the name “Touchard polynomials” appeared and why. Moreover, the name is misleading and brings to confusion in the mathematical community, because it diverts attention away from established publications. Simple professional ethics requires that the article be rewritten and updated. It will be justified to change the title to “exponential polynomials”.
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Ayhan Dil and Veli Kurt, Investigating
Geometric and Exponential Polynomials with Euler-Seidel Matrices, Journal of Integer Sequences, Vol. 14 (2011), Article 11.4.6; Ayhan Dil and Veli Kurt, POLYNOMIALS RELATED TO HARMONIC NUMBERS AND EVALUATION OF HARMONIC NUMBER SERIES I1, INTEGERS 12 (2012),
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Knowledge. I think this name given somehow to the described polynomials is not appropriate. The polynomials described there were used as early as 1843 in the works of Grunert (see , , and ) and possibly could have been used earlier. Bell called them “exponential polynomials”, so did Touchard .
178:. From what I can tellm the two are closely related, but the relationship is not really spelled out. Can someone who knows more a) make the redirect unambiguous (and as specific as possible) and b) take whatever other measures are useful? Should the articles possibly be merged? Thanks! --
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Khristo N. Boyadzhiev, A Series transformation formula and related polynomials, International
Journal of Mathematics and Mathematical Sciences ,Vol. 2005 (2005), Issue 23, Pages 3849-3866.
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Khristo N. Boyadzhiev, Lah numbers, Laguerre polynomials of order negative one, and the nth derivative of exp(1/x), Acta
Universitatis Sapientiae, Mathematica 8(1), (2016), 22–31.
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Khristo N. Boyadzhiev. Exponential polynomials, Stirling numbers, and evaluation of some Gamma integrals, Abstract and
Applied Analysis, Volume 2009, Article ID 168672
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Khristo N. Boyadzhiev, Ayhan Dil, Geometric polynomials: properties and applications to series with zeta values, Analysis
Mathematica, 42 (3) (2016), 203–224
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J. Touchard, “Proprietes arithmetiques de certains nombres recurrents,” Annales de la Soci´et´e
Scientifique de Bruxelles A, vol. 53, pp. 21–31, 1933.
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Khristo N. Boyadzhiev, Power Series with
Binomial Sums and Asymptotic Expansions, Int. Journal of Math. Analysis, Vol. 8 (2014), no. 28, 1389-1414.
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Khristo N. Boyadzhiev, Close encounters with the
Stirling numbers of the second kind, The Mathematics Magazine, 85, No. 4, (October 2012), 252-266.
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J. A. Grunert, “Uber die Summerung der Reihen...,” Journal fur die reine und angewandte Mathematik, vol. 25, pp. 240–279, 1843.
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J. Touchard, “Nombres exponentiels et nombres de Bernoulli,” Canadian Journal of Mathematics, vol. 8, pp. 305–320, 1956.
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L. Carlitz, “Single variable Bell polynomials,” Collectanea Mathematica, vol. 14, pp. 13–25, 1962.
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E. T. Bell, “Exponential numbers,” The American Mathematical Monthly, vol. 41, no. 7, pp. 411–419,
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E. T. Bell, “Exponential polynomials,” Annals of Mathematics, vol. 35, no. 2, pp. 258– 277, 1934.
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