Abstract
In general (see e.g. Cartan in Elementary Theory of Analytic Functions of One or Several Complex Variables, 1963), given two (formal) power series g(x)=b 0+xb 1+⋯+x n b n +⋯ and f(x)=xa 1+⋯+x n a n +⋯ (with a 0=f(0)=0) it is well known that the composition of g with f, in symbols g(f(x)), is a formal power series when coefficients a j and b k are taken in a commutative field. Furthermore, if the constant term a 0 of the power series f is not 0, the existence of the composition g(f(x)) has been an open problem for many years and only recently has received some partial answers (see Gan and Knox in Int. J. Math. Math. Sci. 30:761–770, 2002). The notion of slice-regularity, recently introduced by Gentili and Struppa (Adv. Math. 216:279–301, 2007), for functions in the non-commutative division algebra ℍ of quaternions guarantees their quaternionic analyticity but the non–commutativity of the product in ℍ requires special attention even to define their multiplication (see also Gentili and Stoppato in Michigan Math. J. 56:655–667, 2008). In this paper we face the problem of defining the (slice-regular) composition g⊙f of two slice-regular functions f,g; this turns out to be defined as an extension of the standard composition g∘f of functions in a non-commutative setting which takes into account a non-commutative version of Bell polynomials and a generalization of the Faà di Bruno Formula.
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- 1.
- 2.
A polynomial B n,d is homogeneous of degree d if
$$B_{n,d}(\lambda y_1,\lambda y_2, \ldots, \lambda y_n)=\lambda^d B_{n,d}( y_1, y_2, \ldots, y_n) $$for any λ.
- 3.
A derivation is an endomorphism
of an associative (generally non-commutative) algebra
with unit 1, such that D(y
1⋅y
2)=D(y
1)⋅y
2+y
1⋅D(y
2). In particular D(1)=0.
with unit 1, such that D(y
1⋅y
2)=D(y
1)⋅y
2+y
1⋅D(y
2). In particular D(1)=0.References
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Gan, X.-X., Knox, N.: On composition of formal power series. Int. J. Math. Math. Sci. 30, 761–770 (2002)
Gentili, G., Stoppato, C.: Zeros of regular functions and polynomials of a quaternionic variable. Mich. Math. J. 56, 655–667 (2008)
Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216, 279–301 (2007)
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Gentili, G., Stoppato, C., Struppa, D.C., Vlacci, F.: Recent developments for regular functions of a hypercomplex variable. In: Hypercomplex Analysis. Trends in Mathematics, pp. 165–186. Birkhäuser, Basel (2009)
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Acknowledgements
The author has been partially supported by Progetto MIUR di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse and by G.N.S.A.G.A (gruppo I.N.d.A.M).
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Vlacci, F. (2013). Regular Composition for Slice-Regular Functions of Quaternionic Variable. In: Gentili, G., Sabadini, I., Shapiro, M., Sommen, F., Struppa, D. (eds) Advances in Hypercomplex Analysis. Springer INdAM Series, vol 1. Springer, Milano. https://doi.org/10.1007/978-88-470-2445-8_9
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DOI: https://doi.org/10.1007/978-88-470-2445-8_9
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