On solutions of holonomic divided-difference equations on nonuniform lattices (Q402328)

The authors study polynomials orthogonal on nonuniform lattices NEWLINE\[NEWLINEx(s)=c_1q^s+c_2q^{-s}+c_3 \text{ if }q\not= 1\text{ and } x(s)=c_4s^2+c_5s+c_6 \text{ if }q=1.NEWLINE\]NEWLINE In order to complete the existing characterization of the classical orthogonal polynomials on this type of lattices, they introduce two new monomial bases for the expansion NEWLINE\[NEWLINEP_n(x(s)=\sum_{k=0}^n\,a_k x(s)^k,NEWLINE\]NEWLINE and their formal Stieltjes function NEWLINE\[NEWLINE\int_a^b\,{\text{d}\mu(x(s))\over x(z)-x(s)}=\sum_{n=0}^{\infty}\,{\mu_n\over x(s)^{n+1}};\;\mu_n=\int_a^b\,x(s)^n \text{d}\mu(x(s)),\;x(z)\not\in (a,b).NEWLINE\]NEWLINENEWLINEAs indicated in the introduction, the first basis \(\{F_n\}_n\) is chosen to provide nice operational properties:NEWLINENEWLINEThe basis \(\{F_n(x(s))\}_n\) of polynomials of degree \(n\) in \(x(s)\) satisfy NEWLINENEWLINE\[NEWLINE\mathbf{D}_xF_n(x(s))=a_nnF_{n-1}(x(s)),\;\text\textbf{D}_x{1\over F_n(x(s))}={b_n\over F_{n+1}(x(s))},NEWLINE\]NEWLINE \quad and NEWLINENEWLINE\[NEWLINE\mathbf{S}_xF_n(x(s))=c_nF_n(x)s))+d_nF_{n-1}(x(s)),\;\text\textbf{S}_x{1\over F_n(x(s))}={e_n\over F_{n}(x(s))} + {f_n\over F_{n+1}(x(s))},NEWLINE\]NEWLINE \quad with given constants \(a_n,b_n,c_n,d_n,e_n\) and \(f_n\); the \textbf{companion operators} are given by NEWLINE\[NEWLINE\mathbf{D}_x f(x(s))={f(x(s+{1\over 2}))-f(x(s-{1\over 2}))\over x(s+{1\over 2})-x(s-{1\over 2})},\;\text\textbf{S}_x f(x(s))={f(x(s+{1\over 2}))-f(x(s-{1\over 2}))\over 2}.NEWLINE\]NEWLINE \quad (these operators transform polynomials of degree \(n\) in \(x(s)\) into degree \(n-1\) resp. \(n\) polynomials)NEWLINENEWLINE\vskip0.3cm NEWLINETo achieve solutions of arbitrary linear divided-difference equations with polynomial coefficients involving products of \(\mathbf{D}_x\text{ and \textbf{S}}_x\) only (more suitable for general Askey-Wilson polynomials), the second basis \(\{B_n\}_n\) is introduced on the general \(q\)-quadratic lattice NEWLINE\[NEWLINEx(s)=uq^s+vq^{-s}NEWLINE\]NEWLINE by NEWLINE\[NEWLINEB_n(a,s)=(2 a u q^s,q)_n(2a v q^{-s},q)_n,\;n\geq 1;\;B_0(x,s)=1.NEWLINE\]NEWLINE (the notation \((\cdots; q)_n\) indicates the customary \(q\)-Pochhammer symbol)NEWLINENEWLINEThe elements of this basis satisfy a host of properties and lead to quite a number of explicit series solutions to the divided difference equations studied.NEWLINENEWLINEThe connection between the two bases is given in the paper in Proposition 16: NEWLINE\[NEWLINEF_n(x(s))=\sum_{j=0}^n\,r_{n,j}B_j(a,s),\;B_n(a,s)=\sum_{j=0}^n\,s_{n,j}F_j(x(s)),NEWLINE\]NEWLINE with explicit expressions for the connection coefficients \(r_{n,j}\) and \(s_{n,j}\).NEWLINENEWLINEThe layout of the paper is as follows:NEWLINENEWLINE1. IntroductionNEWLINENEWLINE2. A new basis compatible with the companion operatorsNEWLINENEWLINE3. Algorithmic series solutions of divided-difference equationsNEWLINENEWLINE4. Applications and illustrationsNEWLINENEWLINE5. Conclusions and perspectivesNEWLINENEWLINEReferences (35 items)