Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Special Matrices

Editor-in-Chief: da Fonseca, Carlos Martins

1 Issue per year


Covered by:
WoS (ESCI)
SCOPUS
MathSciNet
zbMATH

CiteScore 2017: 0.29

SCImago Journal Rank (SJR) 2017: 0.295
Source Normalized Impact per Paper (SNIP) 2017: 0.481

Mathematical Citation Quotient (MCQ) 2017: 0.17

Open Access
Online
ISSN
2300-7451
See all formats and pricing
More options …

On the q-Lie group of q-Appell polynomial matrices and related factorizations

Thomas Ernst
Published Online: 2018-03-02 | DOI: https://doi.org/10.1515/spma-2018-0009

Abstract

In the spirit of our earlier paper [10] and Zhang and Wang [16],we introduce the matrix of multiplicative q-Appell polynomials of order M ∈ ℤ. This is the representation of the respective q-Appell polynomials in ke-ke basis. Based on the fact that the q-Appell polynomials form a commutative ring [11], we prove that this set constitutes a q-Lie group with two dual q-multiplications in the sense of [9]. A comparison with earlier results on q-Pascal matrices gives factorizations according to [7], which are specialized to q-Bernoulli and q-Euler polynomials.We also show that the corresponding q-Bernoulli and q-Euler matrices form q-Lie subgroups. In the limit q → 1 we obtain corresponding formulas for Appell polynomial matrices.We conclude by presenting the commutative ring of generalized q-Pascal functional matrices,which operates on all functions f ∈ Cq .

Keywords: q-Lie group; multiplicative q-Appell polynomial matrix; commutative ring; q-Pascal functional matrix

MSC 2010: Primary 17B99; Secondary 17B37; 33C80; 15A23

References

  • [1] L. Aceto and D. Trigiante, The matrices of Pascal and other greats, Amer. Math. Monthly 108 no. 3 (2001), 232-245Google Scholar

  • [2] L. Aceto, H.R. Malonek and G. Tomaz, A unified matrix approach to the representation of Appell polynomials, Integral Transforms Spec. Funct. 26 no. 6 (2015), 426-441Google Scholar

  • [3] T.Arponen, A matrix approach to polynomials, Linear Algebra Appl. 359 (2003), 181-196Google Scholar

  • [4] T. Ernst, A comprehensive treatment of q-calculus, Birkhäuser 2012CrossrefGoogle Scholar

  • [5] T. Ernst, q-Leibniz functional matrices with connections to q-Pascal and q-Stirling matrices. Adv. Studies Contemp. Math. 22 no. 4, (2012), 537-555Google Scholar

  • [6] T. Ernst, q-Pascal and q-Wronskian matrices with implications to q-Appell polynomials. J. discrete math. Article ID 450481, 10 p. (2013)Google Scholar

  • [7] T. Ernst, Faktorisierungen von q-Pascalmatrizen (Factorizations of q-Pascal matrices). Algebras Groups Geom. 31 no. 4, (2014), 387-405Google Scholar

  • [8] T. Ernst, A solid foundation for q-Appell polynomials. ADSA 10, (2015), 27-35Google Scholar

  • [9] T. Ernst, On the q-exponential of matrix q-Lie algebras. Spec. Matrices 5, (2017), 36-50Google Scholar

  • [10] T. Ernst, On several q-special matrices, including the q-Bernoulli and q-Euler matrices. To appear in Linear Algebra Appl.Google Scholar

  • [11] T. Ernst, A new semantics for special functions. Submitted.Google Scholar

  • [12] G.-Y. Lee, J.-S. Kim and S.-G.Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices. Fibonacci Quart. 40, (2002), 203-211Google Scholar

  • [13] Y. Yang and C. Micek, Generalized Pascal functional matrix and its applications. Linear Algebra Appl. 423 no. 2-3, (2007), 230-245Google Scholar

  • [14] Y. Yang and H. Youn, Appell polynomial sequences: a linear algebra approach. J. Algebra Number Theory Appl. 13 no. 1, (2009), 65-98Google Scholar

  • [15] Z. Zhang, The linear algebra of the generalized Pascal matrix. Linear Algebra Appl. 250, (1997), 51-60Google Scholar

  • [16] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties. Discrete Appl. Math. 154 no. 11, (2006), 1622-1632Google Scholar

  • [17] Z. Zhang and X. Wang, A factorization of the symmetric Pascal matrix involving the Fibonacci matrix. Discrete Appl. Math. 155 no. 17, (2007), 2371-2376Google Scholar

About the article

Received: 2017-10-10

Accepted: 2018-02-02

Published Online: 2018-03-02


Citation Information: Special Matrices, Volume 6, Issue 1, Pages 93–109, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2018-0009.

Export Citation

© 2018, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in