Tricomi Continuants

Abstract

In this paper, we introduce and study the Tricomi continuants, a family of tridiagonal determinants forming a Sheffer sequence closely related to the Tricomi polynomials and the Laguerre polynomials. Specifically, we obtain the main umbral operators associated with these continuants and establish some of their basic relations. Then, we obtain a Turan-like inequality, some congruences, some binomial identities (including a Carlitz-like identity), and some relations with the Cayley continuants. Furthermore, we show that the infinite Hankel matrix generated by the Tricomi continuants has an LDU-Sheffer factorization, while the infinite Hankel matrix generated by the shifted Tricomi continuants has an LTU-Sheffer factorization. Finally, by the first factorization, we obtain the linearization formula for the Tricomi continuants and its inverse.

Keywords

• exponential Riordan matrices
• Sheffer polynomials
• umbral operators
• congruences
• binomial identities
• Cayley continuants