×

Matrix extensions of polynomials in several variables. (English) Zbl 1293.15009

E. Erkuş and H. M. Srivastava [Integral Transforms Spec. Funct. 17, No. 4, 267–273 (2006; Zbl 1098.33016)] studied the family of multivariable polynomials generated by \[ \prod_{j=1}^r\left\{\left(1-x_jt^{m_j}\right)^{-\alpha_j}\right\} = \sum_{n=0}^\infty u_n^{(\alpha_1,\dots,\alpha_j)}(x_1,\dots , x_r)t^n, \] where \[ u_n^{(\alpha_1,\dots,\alpha_j)}(x_1,\dots,x_n)=\sum_{m_1k_1+\cdots+m_rk_r=n} (\alpha_1)_{k_1}\dots(\alpha_r)_{k_r} \frac{x_1^{k_1}}{k_1!} \cdots \frac{x_r^{k_r}}{k_r!}. \] When \(m_1=\cdots=m_j=1\), it is the Chan-Chyan-Srivastava multivariable polynomials. When \(m_j=j\) for all \(j\), it is the Lagrange-Hermite polynomials.The notation \[ (\lambda)_k=\lambda(\lambda+1)\cdots(\lambda+k-1)\, \text{ for }\, k=1,2,\dots,\;(\lambda)_0=1 \] is the Pochhammer symbol.
The authors of this paper obtain the matrix version of those multivariable polynomials by defining the Pachhammer symbol of matrices: \[ (A)_k= A(A+1)\cdots(A+(k-1)I)\, \text{ for }\, k=1,2,\dots,\;(A)_0=I. \] Some identities and properties of the polynomials are studied.

MSC:

15A24 Matrix equations and identities
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A60 Functional calculus for linear operators
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 1098.33016
PDFBibTeX XMLCite