The second-order self-associated orthogonal sequences. (English) Zbl 1090.42014

This paper considers orthogonality from the viewpoint of linear forms.
Let \(\mathbf P\) be the linear space of polynomials over \(\mathbb C\) and \(\mathbf P'\) its dual. The action of a dual element on a polynomial is denoted by \[ \langle u,f\rangle\;(u\in\mathbf P',f\in\mathbf P), \] and the well-known calculus of forms (reminiscent of test functions in Schwartz’ space) is introduced.
For an arbitrary sequence of monic polynomials (MPS) \(W_n\) with deg \(W_n=n\), the dual sequence \(w_n\) is introduced by its action \[ \langle w_n,W_m\rangle=\delta_{m,n}\;(\text{Kronecker's delta}), \] and the associated sequence by \[ W_n^{(1)}(x):=\langle w_0,{W_{n+1}(x)-W_{n+1}(\xi)\over x-\xi}\rangle\;(n\geq 0). \]
Higher associated sequences are now defined recursively: \[ W_n^{(r+1)}=\left(W_n^{(r)}\right)^{(1)}\;(n,r\geq 0). \]
A form \(w\in\mathbf P'\) is regular when there exists an MPS \(W_n\) such that \[ \langle w,W_nW_m\rangle =r_n\delta_{n,m}\;(n,m\geq 0;\;r_n\not= 0,\,n\geq 0). \] Here we see that the sequence \(W_n\) is actually a monic orthogonal polynomial sequence (MOPS) with respect to \(w\).
The authors then give a complete solution of the problem to determine all second-order self-associated MOPS with respect to a given form \(w\), i.e., \[ W_n^{(2)}=W_n\;(n\geq 0) \] The solutions depend on three parameters \((\tau,\upsilon,\varepsilon)\) with \(\tau\in\mathbb C,\,\upsilon\in\mathbb C\setminus\{-1,1\}\) and \(\varepsilon^2=1\).
For those sequences the structure relation, second order differential equation and an integral representation are derived. Moreover, they recover as a special case the so-called electrospheric polynomials [A. Guillet, M. Aubert and M. Parodi, Mem. Sci Math. 107, Paris: Gauthier-Villars (1947; Zbl 0039.29801)].


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)


Zbl 0039.29801
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