Erkuş-Duman, Esra Matrix extensions of polynomials in several variables. (English) Zbl 1293.15009 Util. Math. 85, 161-180 (2011). 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. Reviewer: Wai-Shun Cheung (Hong Kong) Cited in 1 ReviewCited in 3 Documents 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.) Keywords:matrix functional calculus; multilinear and multilateral generating functions; Chan-Chyan-Srivastava multivariable polynomials; Lagrange-Hermite polynomials; Pochhammer symbol Citations:Zbl 1098.33016 PDFBibTeX XMLCite \textit{E. Erkuş-Duman}, Util. Math. 85, 161--180 (2011; Zbl 1293.15009)