Recurrence relations for the coefficients in series expansions with respect to semi-classical orthogonal polynomials (Q1430248)

The authors study regular linear functionals on the vector space of all polynomials in one variable with complex coefficients that satisfy \[ D(\varphi L)=\psi L\tag{1,} \] where \(D\) is the differentiation operator and \(\varphi,\,\psi\) are polynomials, deg\(\,\psi\geq 1\). The orthogonal polynomials \(P_k\;(k\geq 0)\) with respect to \(L\) are referred to as semi-classical of class \(s\) when \(s:=\min\max\{\text{deg}\,\varphi -2,\text{deg}\,\psi-1\}\), where the minimum is taken over all pairs of polynomials satisfying (1). The main interest in the paper lies in the Fourier-L-series expansion \[ \sum_k\,a_k[f]P_k;\;a_k[f]:=\langle L,fP_k\rangle\quad (k\geq 0)\tag{2} \] for functions \(f\) that satisfy a differential equation \[ \sum_{j=0}^m\,w_j(x)D^j f(x)=g(x)\tag{3} \] with polynomial coefficients and where \(g\) has an expansion (2). Using basic properties of the orthogonal polynomials, the authors derive a host of recurrence relations for the Fourier-\(L\)-coefficients; a.o. they derive relations between the coefficients for \(f\) and those for \(pF\) (\(p\) an arbitrary polynomial), \((\varphi D+\psi I)f\), etc. The reader is warned, however, that reading this paper is a serious matter: a quite extensive set of special operators plays an important role. The paper is completed by an appendix containing explicit forms for nearly all recurrence relations derived in the previous sections, applied to the case of three different monic, semi-classical sequences of type \(s=1\). 1. Generalized Gegenbauer: \[ \langle L,p\rangle:=C \int_{-1}^1\, (1-x^2)^{\alpha} | x| ^{2\beta +1}p(x)\,dx, \] with \(\alpha,\,\beta>-1\); \(\beta\not= -1/2\), \(C=1/B(\alpha+1,\beta+1)\), \(B(a,b)\) Euler's Beta-function. 2. Bessel-type: \[ \langle L,p\rangle:=\int_{\gamma}\,\rho^{\alpha}(z)p(z)\,dz+\lambda p(0), \] with \(\lambda\) complex, \(2\alpha\not= 0,-1,-2,\ldots\), \(\gamma\) the unit circle and \[ \rho^{\alpha}(z)={1/2 \over 2\pi i}\sum_{k=0}^{\infty} \, {1\over (2\alpha-1)_k} \left( -{2\over z}\right)^k. \] 3. Laguerre-type: \[ \langle L,p\rangle:={1\over \Gamma(\alpha+1)}\int_0^{\infty} \,x^{\alpha}e^{-x}p(x)\,dx+Mp(0), \] with \(\alpha>-1,\;M\geq 0\).