×
Kellner, Bernd C.

On (self-)reciprocal Appell polynomials: symmetry and Faulhaber-type polynomials. (English) Zbl 1490.11033

Integers 21, Paper A119, 19 p. (2021).
Summary: The main purpose of this paper is to study generalized (self-)reciprocal Appell polynomials, which play a certain role in connection with Faulhaber-type polynomials. More precisely, we show for any Appell sequence when satisfying a reflection relation that the Appell polynomials can be described by Faulhaber-type polynomials, which arise from a quadratic variable substitution. Furthermore, the coefficients of the latter polynomials are given by values of derivatives of generalized reciprocal Appell polynomials. Subsequently, we show some applications to the Bernoulli and Euler polynomials. In the context of power sums the results transfer to the classical Faulhaber polynomials.

MSC:

11B83 Special sequences and polynomials
11B68 Bernoulli and Euler numbers and polynomials
PDF BibTeX XML Cite
\textit{B. C. Kellner}, Integers 21, Paper A119, 19 p. (2021; Zbl 1490.11033)
Full Text: arXiv Link

References:

[1] T. Agoh, A new property of Appell sequences and its application, Integers 21 (2021), #A42, 1-16. · Zbl 1483.11035
[2] P. Appell, Sur une classe de polynômes, Ann. Sci.École Norm. Sup. (2) 9 (1880), 119-144. · JFM 12.0342.02
[3] L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, 1974.
[4] A. von Ettingshausen, Vorlesungenüber die höhere Mathematik, vol. 1, Carl Gerold, Vienna, 1827.
[5] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA, USA, 1994.
[6] B. C. Kellner, Faulhaber polynomials and reciprocal Bernoulli polynomials, preprint (2021), 1-35, submitted. arXiv:2105.15025 [math.NT]
[7] B. C. Kellner, Shifted sums of the Bernoulli numbers, reciprocity, and denominators, preprint (2021), 1-11, submitted. arXiv:2105.15049 [math.NT]
[8] D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), 277-294. · Zbl 0797.11026
[9] I. Lah, A new kind of numbers and its application in the actuarial mathematics, Inst. Actuários Portug., Bol. 9 (1954), 7-15. · Zbl 0055.37902
[10] É. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891.
[11] N. E. Nørlund, Vorlesungenüber Differenzenrechnung, J. Springer, Berlin, 1924.
[12] S. Roman, The Umbral Calculus, Academic Press, New York, 1984. · Zbl 0536.33001
[13] G.-C. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
[14] E. Schröder, Eine Verallgemeinerung der Mac-Laurinschen Summenformel, Kantonsschule Zürich, 1867, 1-28.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.