Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation

ABSTRACT

In this paper we will introduce a sequence of complex numbers that are called the Jacobi numbers. This sequence generalizes in a natural way several sequences that are known in the literature, such as Catalan numbers, central binomial numbers, generalized catalan numbers, the coefficient of the Hilbert matrix and others. Subsequently, using a study of the polynomial of Jacobi, we give an evaluation of the Hankel determinants that associated with the sequence of Jacobi numbers. Finally, by finding a relationship between the Jacobi numbers and generalized harmonic numbers, we determine the evaluation of the Hankel determinants that are associated with generalized harmonic numbers.

KEYWORDS:

• Catalan numbers
• Hankel determinant
• orthogonal Jacobi polynomial
• generalized harmonic numbers

MSC:

• 11B83
• 11Y55
• 11C20
• 11B65
• 33C45

Acknowledgments

It is a pleasure to thank the referees of the present paper for their careful reading and very valuable comments. The author thanks Professor Ridha Sfaxi, for his remarks and suggestions to improve the presentation and contents of the manuscript. The author thank the Deanship of Scientific Research and the Deanship of Community Service at Majmaah University, Kingdom of Saudi Arabia, for supporting this work. The author thank Zaid bin Abdullah Al Muraikhi, Abdul Malik bin Abdul Aziz Al Mleifi, Abdul Majeed bin Badr Al Bader, Yazid bin Majid Al Rasheed and Ahmed bin Abdul Karim Al Tayyar for participating in this work.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Wathek Chammam http://orcid.org/0000-0002-3274-8334