On quantum integers and rationals. (English) Zbl 1364.11053
Chamizo, Fernando (ed.) et al., Trends in number theory. Fifth Spanish meeting on number theory, Universidad de Sevilla, Sevilla, Spain, July 8–12, 2013. Proceedings. Providence, RI: American Mathematical Society (AMS); Madrid: Real Sociedad Matemática Española (RSME) (ISBN 978-0-8218-9858-1/pbk; 978-1-4704-2729-0/ebook). Contemporary Mathematics 649, 107-130 (2015).
This paper is aimed to enrich the increasing literature about quantum mathematics with a compact, detailed and well-argumented presentation of several \(q\)-numbers’ properties. Neither elaborated nor ambitious enough to become a manifesto on this subject, it is however a useful and easy to read reference with original results.
After having recalled specific [V. Kac and P. Cheung, Quantum calculus. New York, NY: Springer (2002; Zbl 0986.05001); L. Di Vizio et al., Gaz. Math. Soc. Math. Fr. 96, 20–49 (2003; Zbl 1063.39015); T. Ernst, A comprehensive treatment of \(q\)-calculus. Basel: Birkhäuser (2012; Zbl 1256.33001)] and topic-related [G. E. Andrews, The theory of partitions. Addison-Wesley. 56–57 (1976; Zbl 0371.10001), Chapter 4; R. P. Stanley, Enumerative combinatorics. Vol. 1. 2nd ed. Cambridge: Cambridge University Press (2012; Zbl 1247.05003); A. Borisov et al., J. Number Theory 109, No. 1, 120–135 (2004; Zbl 1063.39020); corrigendum ibid. 145, 632–634 (2014; Zbl 1296.11010); D. M. Bradley, J. Algebra 283, No. 2, 752–798 (2005; Zbl 1114.11075)] works, the authors define the \(q\)-analog of a positive integer \(m\) in an associative ring \(R\) as \[ (m)_q = \sum_{i=0}^{m-1} q^i \in R , \] being \(q \in R\).
Then they illustrate the \(q\)-characteristic and the \(q\)-flatness clarifying, by induction, the link between affine endomorphisms and \(q\)-numbers.
The article continues with \(q\)-combinatorics, suggesting further studies of the \(q\)-binomial coefficients [D. E. Knuth and H. S. Wilf, J. Reine Angew. Math. 396, 212–219 (1989; Zbl 0657.10008)] and about quantum group theory [Ch. Kassel, Quantum groups. New York, NY: Springer-Verlag (1995; Zbl 0808.17003); G. Lusztig, Introduction to quantum groups. Boston, MA: Birkhäuser (2010; Zbl 0788.17010)]; it proves the quantum Lucas formula by induction and, alternatively, via a theorem of V. J. W. Guo and J. Zeng [Eur. J. Comb. 27, No. 6, 884–895 (2006; Zbl 1111.05009)].
The main achievement is the introduction of \(q\)-rational numbers through an admissible choice of roots in the multiplicative monoid of \(R\) and, according to [T. Ernst , J. Nonlinear Math. Phys. 10, No. 4, 487–525 (2003; Zbl 1041.33013)], the subsequent \(q\)-analog of a real number \(r\) defined as \[ (r)_q = \frac{1-q^r}{1-q} , \] if \( q \in {\mathbb R}_{>0}\) is not equal to 1.
Also relevant is the notion of “twisted power” discussed with relation to the Pochhammer symbol, the Artin-Schreier equation [E. Artin and O. Schreier, Abh. Math. Sem. Univ. Hamburg 5, No. 1, 225–231 (1927; JFM 53.0144.01)], the Frobenius identity and, once again, the Lucas theorem [P. Littelmann, J. Am. Math. Soc. 11, No. 3, 551–567 (1998; Zbl 0915.20022)]. If \( x \in A\) and \(n \in \mathbb N\), the \(n\)-th twisted power of \(x\) (with respect to \( \sigma \)) is \[ x^{(n)_{\sigma}} = \prod_{i=0}^{m-1} {\sigma}^i (x), \] being \(A\) a commutative \(R\)-algebra with an endomorphism \( \sigma \).
For the entire collection see [Zbl 1327.11003].
After having recalled specific [V. Kac and P. Cheung, Quantum calculus. New York, NY: Springer (2002; Zbl 0986.05001); L. Di Vizio et al., Gaz. Math. Soc. Math. Fr. 96, 20–49 (2003; Zbl 1063.39015); T. Ernst, A comprehensive treatment of \(q\)-calculus. Basel: Birkhäuser (2012; Zbl 1256.33001)] and topic-related [G. E. Andrews, The theory of partitions. Addison-Wesley. 56–57 (1976; Zbl 0371.10001), Chapter 4; R. P. Stanley, Enumerative combinatorics. Vol. 1. 2nd ed. Cambridge: Cambridge University Press (2012; Zbl 1247.05003); A. Borisov et al., J. Number Theory 109, No. 1, 120–135 (2004; Zbl 1063.39020); corrigendum ibid. 145, 632–634 (2014; Zbl 1296.11010); D. M. Bradley, J. Algebra 283, No. 2, 752–798 (2005; Zbl 1114.11075)] works, the authors define the \(q\)-analog of a positive integer \(m\) in an associative ring \(R\) as \[ (m)_q = \sum_{i=0}^{m-1} q^i \in R , \] being \(q \in R\).
Then they illustrate the \(q\)-characteristic and the \(q\)-flatness clarifying, by induction, the link between affine endomorphisms and \(q\)-numbers.
The article continues with \(q\)-combinatorics, suggesting further studies of the \(q\)-binomial coefficients [D. E. Knuth and H. S. Wilf, J. Reine Angew. Math. 396, 212–219 (1989; Zbl 0657.10008)] and about quantum group theory [Ch. Kassel, Quantum groups. New York, NY: Springer-Verlag (1995; Zbl 0808.17003); G. Lusztig, Introduction to quantum groups. Boston, MA: Birkhäuser (2010; Zbl 0788.17010)]; it proves the quantum Lucas formula by induction and, alternatively, via a theorem of V. J. W. Guo and J. Zeng [Eur. J. Comb. 27, No. 6, 884–895 (2006; Zbl 1111.05009)].
The main achievement is the introduction of \(q\)-rational numbers through an admissible choice of roots in the multiplicative monoid of \(R\) and, according to [T. Ernst , J. Nonlinear Math. Phys. 10, No. 4, 487–525 (2003; Zbl 1041.33013)], the subsequent \(q\)-analog of a real number \(r\) defined as \[ (r)_q = \frac{1-q^r}{1-q} , \] if \( q \in {\mathbb R}_{>0}\) is not equal to 1.
Also relevant is the notion of “twisted power” discussed with relation to the Pochhammer symbol, the Artin-Schreier equation [E. Artin and O. Schreier, Abh. Math. Sem. Univ. Hamburg 5, No. 1, 225–231 (1927; JFM 53.0144.01)], the Frobenius identity and, once again, the Lucas theorem [P. Littelmann, J. Am. Math. Soc. 11, No. 3, 551–567 (1998; Zbl 0915.20022)]. If \( x \in A\) and \(n \in \mathbb N\), the \(n\)-th twisted power of \(x\) (with respect to \( \sigma \)) is \[ x^{(n)_{\sigma}} = \prod_{i=0}^{m-1} {\sigma}^i (x), \] being \(A\) a commutative \(R\)-algebra with an endomorphism \( \sigma \).
For the entire collection see [Zbl 1327.11003].
Reviewer: Enzo Bonacci (Latina)
MSC:
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 05A30 | \(q\)-calculus and related topics |
| 13B10 | Morphisms of commutative rings |
| 13B25 | Polynomials over commutative rings |
