## 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)].

### MSC:

 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
Full Text: