Catalan–Qi Numbers, Series Involving the Catalan–Qi Numbers and a Hankel Determinant Evaluation

Abstract

In this paper, using a study of the polynomial of Jacobi, we give an evaluation of the Hankel determinants that are associated with the sequence of Catalan–Qi numbers and several sequences of series involving the Catalan–Qi numbers.

1. Introduction

In [1], Chammam generalizes several formulas and series identities involving the Catalan numbers and establishes several new formulas and series identities. In this paper, we continue our study of analytic and algebraic properties of Catalan–Qi numbers and series involving the Catalan–Qi numbers by giving a complete evaluation of their Hankel determinants. This type of study generalizes the work of Aloui [2] and Chammam [3, 4] and is based on the algebraic analysis of Jacobi orthogonal polynomials. The studies of these determinants are very important in different scientific disciplines, for example, in physics, especially the resolution of Toda’s equation [5].

2. Notation and Preliminary Results

Let be the linear space of polynomials in one variable with complex coefficients and its topological dual space. We denote by the action of on and by , , the sequence of moments of with respect to the polynomial sequence . We define some operations in . For linear functionals and , any polynomial and any , let , , , and be the linear functionals defined by duality [6, 7]:

A linear functional is called normalized if it satisfies .

With any sequence of complex numbers , we can associate a unique linear functional given by .

The linear functional is said to be regular if the Hankel determinant for every integer . In this case, there exists a unique sequence of monic polynomials (SMPs) ; i.e., in a way that

The sequence is said to be the sequence of monic orthogonal polynomials (SMOPs) with respect to .

Notice that can be represented by the following formula:

The orthogonality of can be characterized by a three-term recurrence relation (TTRR) [6]:

With initial values , where and are sequences of complex numbers such as , and the convention .

Furthermore,

When is a SMOP with respect to a linear functional , then the sequence of monic polynomials , where , is also orthogonal with respect to and satisfies the following TTRR [6, 7]:

With initial values and and where

A linear functional is said to be classical when it is quasidefinite and there exist two polynomials and , monic, , and , such that satisfies Pearson’s equation (see [6, 7]):

In such a case, the corresponding SMOP is said to be classical. Any shift leaves invariant the classical character. Indeed, the shifted linear functional fulfils [6]:where and .

3. Hankel Determinants of Catalan–Qi Numbers

In 2019, Chammam [3] introduced a sequence of complex numbers that are called the Jacobi numbers, which is defined as follows.

Let , with , , , and , :

With any sequence of complex numbers , we can associate a unique linear functional given by . If we consider the form , moments with respect to the sequence are . Equivalently, is monic and satisfies the following functional equation:

While (9) and [6], is a shifted Jacobi; more precisely,

Under the assumptions , , et , , the linear functionals and their shifted are regular. Denoting by , (resp., ) be the SMOP with respect to (resp., ).

In view of (12), the following connection relation is immediate [6, 7]:

From the TTRR satisfied by the Jacobi SMOP , we can obtain the TTRR satisfied by :where

So,where

In the same article, Chammam has determined a complete evaluation of the Hankel determinant associated with the sequence . It proves the following result.

Theorem 1 (see [3], Theorem 1). If , , and are complex numbers such that , , , and , , thenRecall a binomial coefficient is indexed by a pair of integers and is written as , and this coefficient can be generalized to all complex , and by the following formula:

3.1. Application: Hankel Determinant of Catalan–Qi Numbers

For the past few years, the Professor Feng Qi and his coauthors, in defining the Catalan–Qi numbers, have published out important results of this nice number [8–12].

This generalization of the Catalan numbers was introduced according to Euler–Gamma function as follows:where the rising factorials are defined by

If and are complex numbers such that , and for , by substitution in the last theorem, we obtain a completed evaluation of the Hankel determinants that are associated with the sequence of Catalan–Qi numbers:

For and and using the Gamma duplication formula, we obtained

4. Hankel Determinants of Series Involving the Catalan–Qi Numbers

Our second objective, essential although complex, concerns the evaluation of the Hankel determinant of order , associated with the sequence given by

The series involving the Catalan–Qi numbers are a special case, and it is clear [1] that

The sequence satisfies

Using (26), elementary row operations give

From (24), we can writewhere is the right multiplication of by .

Taking into account (28), we can writewhere and

Observe that

It follows that

From the regularity of the linear functional of which its corresponding SMOP is , we findwhere

Substituting (29) into (33), we find

Lemma 1. If , , and are complex numbers such that , , , and , , thenwhereIn the above, is the sequence of associated polynomials of the first kind of the sequence .

Proof. SetSince are normalized with and is monic, then is a sequence of monic polynomials with .
In order to establish a recurrence relation verified by , we need the following formula [6]:Using (39) with and , we getBut, in view of (31),Then,Since is a SMOP with respect to , it comes that ( is the “Kronecker delta”). Obviously, the SMP is orthogonal because it satisfies the following TTRR:While referring to [6], the SMOP is the corecursive of the SMOP since it is generated by the TTRR (16), where is replaced by and in which is replaced by . To this topic, let us recall that the following connection relation holds:This leads to the desired expression of .

Lemma 2. If , , and are complex numbers such that , , , and , , then the following holds:(i).(ii)(iii)If and where

Proof.  (i)By (13), it is clear that . It is well known that satisfies a first structure relation [6]:whereLet in (45) to getThen,And so thatUsing (5) with and , we get for :ButAnd so thatwhereThen, (6) becomesFor , we find this result.
Observe thatThen,And so thatwhere

Using the previous lemmas, we will be able to give the following results.

Theorem 2. For , where , , , and , , we havewithwhere

Proof. From Lemma 2 where , we havewhereBut, on the one hand, by Lemma 1, we haveThen,On the other hand, by Lemma 2, we havewhereConsequently,Using the binomial coefficient, we obtained the result.

4.1. Application: Hankel Determinant of Series Involving the Catalan–Qi Numbers

For , by substitution in the last theorem, we obtain a completed evaluation of the Hankel determinants that are associated with the sequence of series involving the Catalan–Qi numbers.

Proposition 1. If and are complex numbers such that , we havewhereFor and and using the Gamma duplication formula, we obtained

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under project no. 1439-12.