[Paper Review] A characterization theorem and its applications for d-orthogonality of Sheffer polynomial sets

Serhan Varma|arXiv (Cornell University)|Mar 23, 2016

Mathematical functions and polynomials16 references3 citations

TL;DR

This paper presents a characterization theorem for Sheffer polynomial sets that are d-orthogonal, deriving their generating functions and identifying conditions under which such d-orthogonal Sheffer polynomials exist. The key contribution is a general generating function form (Theorem 2.2) enabling the discovery of new d-orthogonal polynomial sets and explicit construction of their d-dimensional functional vectors.

ABSTRACT

The purpose of this paper is to find the characterization of the Sheffer polynomial sets satisfying the d-orthogonality conditions. The generating function form of these polynomial sets is given in Theorem 2.2. As applications of the Theorem 2.2, we revisit the d-orthogonal polynomial sets exist in the literature and discover new d-orthogonal polynomial sets. Moreover, we obtain the d-dimensional functional vector ensuring the d-orthogonality of these new polynomial sets.

Motivation & Objective

• To characterize Sheffer polynomial sets that satisfy d-orthogonality conditions using generating functions.
• To extend known results on d-orthogonal polynomials to the Sheffer class, generalizing classical orthogonal polynomials like Hermite, Laguerre, and Meixner.
• To derive new d-orthogonal polynomial sets by applying the characterization theorem.
• To explicitly construct the d-dimensional functional vector ensuring d-orthogonality for newly identified polynomial sets.
• To unify and generalize existing results on d-orthogonality within the Sheffer framework, particularly for d ≥ 2.

Proposed method

• Derives a generating function form for d-orthogonal Sheffer polynomial sets via Theorem 2.2, based on the functional relations between A(t) and H(t).
• Applies the d-orthogonality condition: ⟨u_k, P_n P_m⟩ = 0 for m > nd + k, with non-vanishing inner product at m = nd + k.
• Uses the recurrence relation of order d+1: xP_n(x) = ∑_{k=0}^{d+1} α_{k,d}(n) P_{n−d+k}(x), with regularity conditions on coefficients.
• Employs generating function techniques and functional calculus involving the compositional inverse of H(t) and the operator σ = H*(Δ).
• Applies Lemma 2.5 to derive lowering operators and express the linear functionals ⟨u_r, f⟩ via finite differences and Pochhammer symbols.
• Constructs the d-dimensional functional vector using the formula ⟨u_r, f⟩ = 1/r! [σ^r / A(σ)] f(x)|_{x=0}, involving binomial and hypergeometric-type series.

Experimental results

Research questions

• RQ1 What conditions on A(t) and H(t) ensure that a Sheffer polynomial set is d-orthogonal?
• RQ2 Can the characterization of d-orthogonal Sheffer polynomials be generalized beyond d=1 and d=2?
• RQ3 What new d-orthogonal polynomial sets can be derived from the proposed generating function form?
• RQ4 How can the d-dimensional functional vector be explicitly constructed for such d-orthogonal Sheffer sets?
• RQ5 Do known d-orthogonal polynomials (e.g., Meixner-type) fit into this generalized Sheffer framework?

Key findings

• Theorem 2.2 provides a complete generating function characterization for d-orthogonal Sheffer polynomial sets, generalizing classical results for d=1.
• New d-orthogonal polynomial sets are derived, including a Meixner-type family for d ≥ 2 with generating function e^{π_{d-2}(t)}(1−t)^{-β}(1 + (c−1)/(2c) · (t²−2t)/(1−t)²)^x.
• For d ≥ 2, the resulting polynomial sets are not orthogonal (since d > 1), but are d-orthogonal with respect to a d-dimensional functional vector.
• The d-dimensional functional vector is explicitly constructed as ⟨u_r, f⟩ = 1/r! ∑_{i=0}^r ∑_{j=0}^∞ [∑_{k=0}^∞ (β + i/d)_{k+j} (−1)^k (dc/(1−c))^k / k! ] (dc/(1−c))^j f(j)/j!.
• The functional vector expression involves Pochhammer symbols and finite differences, with a closed-form representation via binomial and hypergeometric-type series.
• For d=1, the result reduces to the classical Meixner polynomials, confirming consistency with known orthogonal cases.