Geodesic
Specific values of partial Bell polynomials and series expansions for real powers of functions and for composite functions
Filomat, Tome 37 (2023) no. 28, p. 9469

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

DOI

Starting from Maclaurin’s series expansions for positive integer powers of analytic functions, the authors derive an explicit formula for specific values of partial Bell polynomials, present a general term of Maclaurin’s series expansions for real powers of analytic functions, obtain Maclaurin’s series expansions of some composite functions, recover Maclaurin’s series expansions for real powers of inverse sine function and sinc function, recover a combinatorial identity involving the falling factorials and the Stirling numbers of the second kind, deduce an explicit formula of the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, recover an explicit formula of the Bernoulli numbers in terms of the Stirling numbers of the second kind, recover an explicit formula of the Bell numbers in terms of the Stirling numbers of the second kind, reformulate three specific partial Bell polynomials in terms of central factorial numbers of the second kind, and present some Maclaurin’s series expansions and identities related to the Euler numbers and their generating function.
DOI : 10.2298/FIL2328469Q
Classification : 41A58, 11B68, 11B73, 11B83, 11C08, 33B10
Keywords: Maclaurin’s series expansion, partial Bell polynomial, Bell number, Bernoulli number, Stirling number, positive integer power, real power, composite function, inverse sine function, sinc function, explicit formula, combinatorial identity, central factorial number of the second kind, Euler number 
Feng Qi; Gradimir V Milovanović; Dongkyu Lim. Specific values of partial Bell polynomials and series expansions for real powers of functions and for composite functions. Filomat, Tome 37 (2023) no. 28, p. 9469 . doi: 10.2298/FIL2328469Q
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     author = {Feng Qi and Gradimir V Milovanovi\'c and Dongkyu Lim},
     title = {Specific values of partial {Bell} polynomials and series expansions for real powers of functions and for composite functions},
     journal = {Filomat},
     pages = {9469 },
     year = {2023},
     volume = {37},
     number = {28},
     doi = {10.2298/FIL2328469Q},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2328469Q/}
}
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