On the theory of biorthogonal polynomials

Abstract

Let φ ( x , μ ) \varphi (x,\,\mu ) be a distribution in x ∈ R x \in {\mathbf {R}} for every μ \mu in a real parameter set Ω \Omega . Subject to additional technical conditions, we study m m th degree monic polynomials p m {p_m} that satisfy the biorthogonality conditions \[ ∫ − ∞ ∞ p m ( x ) d φ ( x , μ l ) = 0 , l = 1 , 2 , … , m , m ⩾ 1 \int _{ - \infty }^\infty {{p_m}(x)\,d\varphi (x,{\mu _l}) = 0,} \qquad l = 1,\,2, \ldots ,\,m,\;m \geqslant 1 \] , for a distinct sequence μ 1 , μ 2 , … ∈ Ω {\mu _1},\,{\mu _2},\, \ldots \; \in \Omega \, . Necessary and sufficient conditions for existence and uniqueness are established, as well as explicit determinantal and integral representations. We also consider loci of zeros, existence of Rodrigues-type formulae and reducibility to standard orthogonality. The paper is accompanied by several examples of biorthogonal systems.