The Hankel transform of generalized central trinomial coefficients and related sequences

Abstract

In this paper, we explore the connection between the Hankel transform, Riordan arrays and orthogonal polynomials. For this purpose, we evaluate the Hankel transform of generalized trinomial coefficients, as a closed-form expression, using the method based on the orthogonal polynomials. Since the generalized trinomial coefficients are generalization of several integer sequences, obtained expression is also applicable in these cases. We also showed that the coefficient array of corresponding orthogonal polynomials can be represented in terms of Riordan arrays, which provides the LDL^T decomposition of the Hankel matrix. Moreover, we consider the row sums of the inverse of coefficient array matrix and evaluate its Hankel transform.

Keywords:

• Hankel transform
• orthogonal polynomials
• central trinomial coefficients
• Riordan arrays

AMS Subject Classification :

• Primary: 11B83
• Secondary: 05A19
• 33C45
• 11B37

Acknowledgements

The authors gratefully acknowledge the support from the research projects 144011 and 144023 of the Serbian Ministry of Science.