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

• Main Author: Ernst Thomas
• Format: Article
• Language: English
• Published: De Gruyter 2018-03-01
• Series: Special Matrices
• Subjects: q-lie group; multiplicative q-appell polynomial matrix; commutative ring; q-pascal functional matrix; primary 17b99; secondary 17b37; 33c80; 15a23
• Online Access: https://doi.org/10.1515/spma-2018-0009

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 ∈ C∞q .