Casorati type determinants of some \(\mathfrak {q}\)-classical orthogonal polynomials (Q2790272)

This paper studies the extension of ``symmetry'' of Wronskian and Casorati determinants to the case where the entries are polynomials belonging to the \(q\)-Askey scheme (a good source for these polynomials is Roelof Koekoek, Peter A. Leskey and René F. Swarttouw [\textit{R. Koekoek} et al., Hypergeometric orthogonal polynomials and their \(q\)-analogues. With a foreword by Tom H. Koornwinder. Springer Monographs in Mathematics. Berlin: Springer (2010; Zbl 1200.33012)]).NEWLINENEWLINEThe main result uses the big \(q\)-Jacobi polynomials (\(q\) is real and \(q\not= 1\)), defined using the following normalization NEWLINE\[NEWLINEP_n^{a,b,c,q}(x)={q^{n^2}(aq,cq;q)_n\over (abq^{n+1},q;q)_n}\,{}_3\varphi_2\left( \begin{matrix} q^{-n},abq^{n+1},x \cr aq,cq \end{matrix}\;;q,q\right),NEWLINE\]NEWLINE and defines the \(q\)-Casorati big \(q\)-Jacobi determinant by NEWLINE\[NEWLINE{\mathcal P}_{n,m,x}^{a,b,c;q}=x^{m\choose 2}\text{det}\,(P_{m+j-1}^{a,b,c,q}(x/q^{i-1}))_{i,j=1}^n.NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe main result of the paper isNEWLINENEWLINE{ Theorem 1.1.} For \(n,m\geq 0\) and \(a\not= 0,q\not= 1\), there holds NEWLINE\[NEWLINE{\mathcal P}_{n,m,x}^{a,b,c;q}=(-1)^{nm} q^{mn^2+nm^2-mn} {\mathcal P}_{m,n,x}^{aq^{n+m},bq^{n+m},cq^{n+m};1/q}.NEWLINE\]NEWLINE (It is pointed out in the paper that the determinant in the left-hand side is of size \(n\times n\) and that in the right hand side of size \(m\times m\).)NEWLINENEWLINEThe paper then gives preliminaries and proof in the Sections 2--4 (an important role is played by their Theorem 2.1, taken from the paper by the first author in [the first author, J. Comb. Theory, Ser. A 124, 57--96 (2014; Zbl 1292.33013)]).NEWLINENEWLINENEWLINEFinally, in Section 5, limits and other relations from the \(q\)-Askey scheme are used to find results as given above for aNEWLINENEWLINENEWLINE\(q\)-Casorati \(q\)-Meixner determinant with elements NEWLINE\[NEWLINEM_n^{a,d;q}(q^{-x})=\lim_{c\rightarrow\infty}\,P_n^{a,c/(ad),c,q}(q^{-x});NEWLINE\]NEWLINENEWLINENEWLINENEWLINE\(q\)-Casorati \(q\)-Charlier determinant with elements NEWLINE\[NEWLINEC_n^{d;q}(x)=M_n^{0,d;q}(x);NEWLINE\]NEWLINENEWLINENEWLINENEWLINE\(q\)-Casorati \(q\)-Laguerre determinant with elements NEWLINE\[NEWLINEL_n^{(\alpha)}(x;q)={(q^{\alpha+1};q)_n\over (q;q)_n}\,\lim_{c\rightarrow\infty}\,M_n(cq^{\alpha}x,q^{\alpha},c;q).NEWLINE\]