Research Article | Open Access
Wathek Chammam, "Catalan–Qi Numbers, Series Involving the Catalan–Qi Numbers and a Hankel Determinant Evaluation", Journal of Mathematics, vol. 2020, Article ID 8101725, 8 pages, 2020. https://doi.org/10.1155/2020/8101725
Catalan–Qi Numbers, Series Involving the Catalan–Qi Numbers and a Hankel Determinant Evaluation
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.
In , 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  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 .
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) :
With initial values , where and are sequences of complex numbers such as , and the convention .
With initial values and and where
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 :where and .
3. Hankel Determinants of Catalan–Qi Numbers
In 2019, Chammam  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:
Under the assumptions , , et , , the linear functionals and their shifted are regular. Denoting by , (resp., ) be the SMOP with respect to (resp., ).
From the TTRR satisfied by the Jacobi SMOP , we can obtain the TTRR satisfied by :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 , 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
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  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
It follows that
From the regularity of the linear functional of which its corresponding SMOP is , we findwhere
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 :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 , 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 :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
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
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