2023/10/18 by Hamza Chaggara, Chaggara, Hamza, Abdelhamid Gahami +1 · 1 citation
Mathematics · Computer Science · #Mathematical functions and polynomials #Coding theory and cryptography #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2310.11734
The Brenke type generating functions are the polynomial generating functions of the form ∑n=0∞Pn(x )\over n!tn=A(t)B(xt), where A and B are two formal power series subject to the conditions A(0) B(k)(0)≠0, k=0,1,2…. In this work, we determine all Brenke-type polynomials when they are also 2-orthogonal polynomial sets, that is to say, polynomials satisfying one standard four-term recurrence relation. That allows us, on one hand, to obtain new 2-orthogonal sequences generalizing known orthogonal families of polynomials, and on the other hand, to recover particular cases of polynomial sequences discovered in the context of d-orthogonality. The classification is based on the resolution of a three-order difference equation induced by the four-term recurrence relation satisfied by the considered polynomials. This study is motivated by the work of Chihara who gave all pairs (A (t), B(t)) for which \Pn(x)\n≥ 0 is an orthogonal polynomial sequence.