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Basrah Journal of Science Vol., 37 (1), 25-43, 2019
25
The Operator and the Polynomials
Husam L. Saad Sadeq M. Khalaf
Department of Mathematics, College of Science, Basrah University, Basra, Iraq
[email protected] [email protected]
Doi 10.29072/basjs.20190103
Abstract
Based on basic hypergeometric series, a new generalized -operator has been constructed and
obtained some operator identities. Also, a new polynomial is
introduced. The generating function and its extension, Mehler’s formula and its extension and the
Rogers formula for the polynomials have been achieved by using
the operator . In fact, this work can be considered as a generalization of Liu work’s by imposing
some special values of the parameters in our results. Therefore the -Rogers-Szegö polynomials
can be deduced directly.
Keywords: -operator, generating function, Mehler’s formula, Rogers formula, the -Rogers-
Szegö polynomials.
AMS Classification: 05A30, 33D45
1. Introduction Throught this paper, the notations in [2] will be used here and assuming that .
Definition 1.1. [2] . Let be a complex variable. The -shifted factorial is defined by
∏
∏
The compact notation for the multiple -shifted factorial will be adopted here
where is an integer or .
Definition 1.2. [2] . The basic hypergeometric series is defined by
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
(
)
∑
* ( )+
where ; ; and none of the denominator factors evaluate to zero. The
above series is absolutely convergent for all if , for if and for
if .
Definition 1.3. [2] . The -binomial coefficient is defined by
[ ] {
where .
The following equations will be used in this paper [2]:
(
)
(
) (
) (
)
(
) (
) (
)
where and are integers. Cauchy identity is given by [2]
∑
The special case of Cauchy identity was founded by Euler [2] which is
∑
( )
Definition 1.4. [3] . The operator is defined by
{ }
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27
Theorem 1.5. [3]. (Leibniz rule for ). Let be defined as in (1.8), then
{ } ∑
* + { } { }
The following identities are easy to prove:
Theorem 1.6. [4, 5, 6] . Let be defined as in (1.8), then
{ }
(
)
{ }
{
} ( )
In 1998, Chen and Liu [4] defined the -exponential operator as follows:
Definition 1.7. [4] . The -exponential operator is defined as follows:
∑
( )
Chen and Liu proved the following result:
Theorem 1.8. [4] . Let be defined as in (1.13), then
{ }
{ }
They used the -exponential operator to present an extension for the Askey beta integral.
In 2006, Zhang and Liu [6] used to prove the following result:
Theorem 1.9. [6] . Let be defined as in (1.13), then
{ } (
)
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
In 2007, Fang [7] defined the Cauchy operator (
) as follows:
Definition 1.10. [7] . The Cauchy operator (
) is defined by
(
) ∑
Fang proved the following result:
Theorem 1.11. [7] . Let (
) be defined as in (1.17), then
(
) { }
Fang used Cauchy operator (
) to obtain an extension for the
-Chu-Vandermonde identity. In 2010, Zhang and Yang [8] introduced the finite -exponential operator with two parameters
[
] as follows:
Definition 1.10. [8] . The finite -exponential operator [
] is defined by
[
] ∑
By using this operator, , Zhang and Yang found an extension for -Chu-Vandermonde summation
formula.
In 2010, Liu [1] defined the -Rogers-Szegö polynomial as follows:
Definition 1.12. [1] . The -Rogers-Szegö polynomial is defined by
∑
[ ]
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29
Liu used the -difference equation to prove the following:
Theorem 1.13. [1] . Let be defined as in (1.19), then
• The generating function for
∑
( )
• Mehler’s formula for
∑
( )
provided that
This paper is organized as follows: In section 2, a generalized -operator (
)
and some of its identities will be definded and studied. In section 3, we define a polynomial
and represent it by the operator . The generating function and
its extension for is obtained. In section 4, the Mehler’s formula and
its extension for is derived . while, in section 5, the Rogers formula
for is constructed. Finally, section 6 is focused on the summary of
the results and the conclusions.
2. The Operator and it’s Identities
In this section, we define the generalized -operator (
) as follows:
Definition 2.1. The generalized -operator (
) is defined by
(
) ∑
* (
)+
When , we get the -exponential operator defined by Chen and Liu [4] in
1998. Also when , , we obtain the -exponential operator (
)
defined by Fang [7] in 2007. And when , , , we obtain the
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
finite -exponential operator with two parameters [
] defined by Zhang and Yang
[8] in 2010. Finally, when , , , , we get the generalized
-exponential operator with three parameters [
] defined by Li and Tan [9] in 2016.
In this paper, we will denote to
by . Then the generalized -operator
can be written as follows:
(
) ∑
* (
)+
Theorem 2.2. Let (
) be defined as in (2.2), then
(
) { } ∑
∑
* ( )+
[
( )]
Proof. From the definition of the operator (
) and by using Leibniz rule
(1.9), we have
(
) { }
∑
* (
)+
{ }
∑
* (
)+
∑
{ } { }
∑
* (
)+
∑
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Basrah Journal of Science Vol., 37 (1), 25-43, 2019
31
∑
* (
)+
∑
∑
∑
* (
)+
[ ( )]
∑
∑
* (
)+
[
( )]
∎
By setting in (2.3), we get Theorem 2.11. obtained in Chen and Liu [4] (equation
(1.15)).
Putting in (2.3), we get the following corollary:
Corollary 1. Let (
) be defined as in (2.2), then
(
) { } ∑
* (
)+
Setting in (2.4), we get Theorem 2.9. obtained by Chen and Liu [4] (equation
(1.14)). Setting and in (2.4), we get Theorem 1.3. obtained by Fang [7] (equation
(1.18)).
Theorem 2.3. Let (
) be defined as in (2.2) and , then
(
) { } ∑
∑
* (
)+
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
[
( )]
Proof. From (2.2), we have
(
) { } ∑
* (
)+
{ }
By using Leibniz rule (1.9), we have
(
) { }
∑
* (
)+
∑
{ } { }
∑
* (
)+
∑
{ }
∑
* ( )+
∑
∑
∑
[ (
)]
∑
∑
* (
)+
[
( )]
∎
Setting in (2.5), we get Corollary 2.4. obtained in Zhang and Liu [6] (equation(1.16)).
3. The Generating Function for
In this section we define a polynomial . By using the operator
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Basrah Journal of Science Vol., 37 (1), 25-43, 2019
33
(
), we get the generating function and its extension for the polynomials .
We give some special values to the parameters in the generating function and its extension for
to obtain the generating function and its extension for the
-Rogers-Szegö polynomials .
Definition 3.1. The polynomial is defined by
∑
* +
* ( )+
where
.
Setting , , in (3.1), we get the -Rogers-Szegö polynomials
(2.12) defined by Liu [1] (equation (1.19)).
Theorem 3.2. Let the polynomials be defined as in (3.1), then
(
) { }
Proof.
(
) { }
∑
* (
)+
{ }
∑
* (
)+
{ }
∑
* (
)+
( )
∑
* +
* ( )+
∎
Theorem 3.3. (The generating function for ). Let be defined
as in (3.2), then
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
∑
( )
(
)
provided that the series is absolutely convergent .
Proof.
∑
( )
∑
(
) { } ( )
(
) {∑
( )
}
(
) { }
∑
* (
)+
(
)
∎
Setting , , in (3.3) we obtain the generating function for the polynomials
(2.13) obtained by Liu [1] (equation (1.20)).
Theorem 3.4. (Extension of the generating function for ).
Let be defined as in (3.2), then
∑
( )
∑
∑
[
( )]
*
( )+
Proof.
∑
( )
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Basrah Journal of Science Vol., 37 (1), 25-43, 2019
35
∑
(
) { } ( )
(
) { ∑
( )
}
(
) { }
∑
∑
[
( )]
[ ]
∎
Setting , , in (3.4) we obtain an extension of the generating function for
the polynomials as follows:
∑
( )
∑
∑
(
)
∑
∑
(
)
∑
4. Mehler’s Formula for In the section, we will derive Mehler’s formula and its extension for the polynomials by using
the operator . We give some special values to the parameters in the Mehler’s formula and its
extension for to obtain Mehler’s formula and its extension for the -Rogers-Szegö
polynomials .
Theorem 4.1. (Mehler’s formula for ). Let be defined as in
(3.2), then
∑
( )
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
∑
* (
)+
∑
∑
[ ( )]
* (
)+
provided that .
Proof.
∑
( )
∑
(
) { } ( )
(
) {∑
( )
}
(
) { (
)}
(
) { ∑
* (
)+
}
∑
* (
)+
(
) { }
∑
* (
)+
∑
∑
[ ( )]
* (
)+
∎
Setting , , , , and in equation (4.1) we get Mehler’s
formula for the polynomials (2.14) obtained by Liu [1] (equation (1.21)) as we see in
the following corollary:
Corollary 2. (Mehler’s formula for ). Let
be defined as in (1.19), then
∑
( )
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Basrah Journal of Science Vol., 37 (1), 25-43, 2019
37
provided that
Proof. Setting , , , , and in equation (4.1) we get
∑
( )
∑
(
) ∑
∑
(
)
∑
(
) ∑
( )
∑
( )
∑
(
)
∑
∑
(
)
∎
Theorem 4.2. (Extension of Mehler’s formula for ). Let be
defined as in (3.1), then
∑
(
)
∑
∑
[
( )]
* (
)+
∑
∑
* ( )+
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
*
( )+
Proof.
∑
( )
∑
(
) { }
( )
(
) {∑
( )
}
(
) { ∑
∑
[ ( )]
* (
)+
} ( )
∑
∑
[
( )]
*
( )+
(
) { }
∑
∑
[
( )]
* (
)+
∑
∑
* ( )+
* (
)+
∎
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39
Setting , , , and in equation (4.2) we get an extension of
Mehler’s formula for the polynomials as we see in the following corollary:
Corollary 3. (Extension of Mehler’s formula for ). Let
be defined as
in (1.19), then
∑
( )
∑
∑
Proof.
Setting , , , and in equation (4.2) we get
∑
( )
∑
∑
(
)
∑
∑
(
)
∑
∑
(
)
∑
∑
(
)
∑
∑
( )
∎
5. Rogers Formula for We will derive, in this section, Roger’s formula for the polynomials by using the operator
. We give some special values to the parameters in Rogers formula for
to obtain Rogers formula for the -Rogers-Szegö polynomials
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H. L. Saad and S. M. Khalaf The Operator and the Polynomials
.
Theorem 5.1. (Rogers formula for ). Let be defined as in (3.2),
then
∑
∑
( )
( )
∑
∑
[
( )]
[ ]
provided that .
Proof.
∑
∑
( )
( )
∑
∑
(
) { } ( )
( )
(
) {∑
( )
∑
( )
}
(
) { }
∑
∑
[
( )]
* ( )+
∎
Setting , and in equation (5.1) we obtain Rogers formula for the
polynomials as we see in the following corollary:
Corollary 4. (Rogers formula for ). Let
be defined as in (1.19), then
∑
∑
( )
( )
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41
provided that .
Proof.
Setting , and in equation (5.1) we get
∑
∑
( )
( )
∑
∑
(
)
∑
∑
(
)
∎
6. Conclusions
This paper devoted to study a new generalized -operator (
). Also, a new
polynomial is constructed. The generating function and its
extension for is studied. Also, the Mehler’s formula and its
extension for is investigated. While, the Rogers formula for
is constructed. In order to explore the results, one can imposing
some special values of the parameters. So, by setting , , in the generating
function and its extension for ), the generating function and its
extension for the -Rogers-Szegö polynomials is obtained directly. Also, by
setting , , in Mehler’s formula and its extension for
, the Mehler’s formula and its extension for the polynomials
is achieved directly. Finaly, by setting , , in Rogers formula
for , the Rogers formula for the polynomials is
created.
References
[1] Z.-G. Liu, Two -difference equations and -operator identities, J. Difference Equ. Appl. 16
(2010) 1293–1307.
[2] G. Gasper, M. Rahman, Basic Hypergeometric Series, ed., Cambridge University Press,
Page 18
H. L. Saad and S. M. Khalaf The Operator and the Polynomials
Cambridge, MA, 2004.
[3] S. Roman, More on the umbral calculus, with emphasis on the -umbral caculus, J. Math. Anal.
Appl. 107 (1985) 222–254.
[4] W.Y.C. Chen, Z.G. Liu, Parameter augmentation for basic hypergeometric series, I,
Mathematical Essays in Honor of Gian-Carlo Rota, Eds., B.E. Sagan and R.P. Stanley, Birkhäuser,
Boston, (1998) 111-129.
[5] Z. Z. Zhang, J. Wang, Two operator identities and their applications to terminating basic
hypergeometric series and -integrals, J. Math. Anal. Appl. 312 (2005) 653–665.
[6] Z. Z. Zhang, M. Liu, Application of Operator identities to the multiple -binomial theorem and
-Gauss summation theorem, Discrete Math. 306 (2006) 1424 - 1437.
[7] J.P. Fang, -Differential operator identities and applications, J. Math. Anal. Appl. 332 (2007)
1393-1407.
[8] Z. Z. Zhang, J. Z. Yang, Finite -exponential operators with two parameters and their
applications, ACTA MATHEMATICA SINICA, Chinese Series 53 (2010) 1007–1018. (in
Chinese)
[9] N. N. Li, W. Tan, Two generalized -exponential operators and their applications, Advances in
difference equations 53 (2016) 1-14.
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Basrah Journal of Science Vol., 37 (1), 25-43, 2019
43
وهتعذدة الحذود الوؤثر
حسام لوتي سعذ, صادق هاجذ خلف
، اث ، كليت العلوم ، جاهعت البصرةقسن الرياضي
البصرة ، العراق
الوستلخص
. وحصهنا عهى بعط انًتطابقاث نهًؤثز انعاو -قًنا بتعزيف انًىثزباستخذاو تعزيف انذانت انهنذسيت انفىقيت االساسيت،
صيغت جذنا انذانت انًىنذة وتىسيعها،و. يتعذدة حذود جذيذة عزفناأيضاً،
، يًكن حقيقتفي ان . باستخذاو انًؤثز ًتعذدة انحذود نRogers صيغت و وتىسيعها
انحصىل عهى [ عن طزيق فزض بعط انقيى انخاصت نهًعهًاث في نتائجنا. نذنك يًكن 1]Liu م عًناعتبار هذا انعًم بًثابت تعًيى
-سيجى -يتعذداث حذود روجزس يباشزة. (
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