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)].