Abstract and Applied AnalysisVolume 2012 (2012), Article ID 190726, 17 pagesdoi:10.1155/2012/190726

Research Article

Functions Induced by Iterated Deformed Laguerre Derivative: Analytical and Operational Approach

Sladjana D. Marinković,¹ Miomir S. Stanković,² and Predrag M. Rajković³

¹Department of Mathematics, Faculty of Electronic Engineering, University of Niš, 18000 Niš, Serbia²Department of Mathematics, Faculty of Occupational Safety, University of Niš, 18000 Niš, Serbia³Department of Mathematics, Faculty of Mechanical Engineering, University of Niš, 18000 Niš, Serbia

Received 27 December 2011; Accepted 23 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Sladjana D. Marinković et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The one-parameter deformed exponential function was introduced as a frame that enchases a few known functions of this type. Such deformation requires the corresponding deformed operations (addition and subtraction) and deformed operators (derivative and antiderivative). In this paper, we will demonstrate this theory in researching of some functions defined by iterated deformed Laguerre operator. We study their properties, such as representation, orthogonality, generating function, differential and difference equation, and addition and summation formulas. Also, we consider these functions by the operational method.

1. Introduction

The several parametric generalizations and deformations of the exponential function have been proposed recently in different contexts such as nonextensive statistical mechanics [1, 2], relativistic statistical mechanics [3, 4], and quantum group theory [5–7].

The areas of deformations of the exponential functions have been treated basically along three (complementary) directions: formal mathematical developments, observation of consistent concordance with experimental (or natural) behavior, and theoretical physical developments.

In paper [8], a deformed exponential function of two variables depending on a real parameter is introduced to express discrete and continual behavior by the same. In this function, well-known generalizations and deformations can be viewed as the special cases [1, 5]. Also, its usage can be seen in [9].

In continuation of our previous considerations of the deformed exponential function, we will use its convenience to introduce and research a class of functions of two variables that can be viewed as one-parameter analog of Laguerre polynomials.

The paper is organized as follows. In Section 2, we introduce the deformed exponential function and the related deformation of variable, addition and subtraction. In Section 3, we consider some known and new difference and differential operators, convenient for the work with deformed variables and exponentials. In this environment, in Section 4, we define a class of two-variable functions, called the deformed Laguerre polynomials, by an iterated generalized differential operator. Section 5 is devoted to various properties of these functions, such as orthogonality, summation and addition formulas, differential equations, generating function. Finally, in Section 6 we prove a few operational identities involving the introduced deformed Laguerre polynomials.

Because of the presence of “logarithmic scale,” deformed Laguerre operator and deformed Laguerre polynomials could be suitable for use in control engineering, population dynamics, mathematical modeling of viscous fluids, and oscillating problems in mechanics, like the usual Laguerre operator and Laguerre polynomials that are already used (see [10–12]).

2. The Deformed Exponential Functions

In this section we will present a deformation of an exponential function of two variables depending on parameter , which is introduced in [8].

Let us define function by

Since this function can be viewed as a one-parameter deformation of the exponential function of two variables.

If and , function (2.1) becomes that is, , where is Tsallis -exponential function [1] defined by

If and , function (2.1) becomes that is, a function considered for a generalization of the standard exponential function in the context of quantum group formalism [13].

Notice that function (2.1) can be written in the form Hence, similar to what in [14], we can use cylinder transformation as deformation function by Thus, the following holds:

We can show that function (2.1) holds on some basic properties of the exponential function.

Proposition 2.1. For and , the following holds:

Notice that the additional property is true with respect to the second variable only. However, with respect to the first variable, the following holds:

This equality suggests introducing a generalization of the sum operation Such generalized addition operator was considered in some papers and books (see, e.g., [2] or [14]). This operation is commutative and associative, and zero is its neutral. For , the -inverse exists as and is valid. Hence, is an abelian group, where for or for . In this way, the -subtraction can be defined by

With respect to (2.7), we can prove the next equality for :

Proposition 2.2. For and , the following is valid:

In order to find the expansions of the introduced deformed exponential function, we introduce the generalized backward integer power given by

Proposition 2.3. For function , the following representation holds:

Remark 2.4. Notice that in expressions (2.8) and the first expansion in (2.17) the deformation of variable appears, but, contrary to that in the second expansion in (2.17), the deformation of powers of is present.

3. The Deformed Operators

Let us recall that the -difference operator is

Proposition 3.1 (see [8]). The function is the eigenfunction of difference operator with eigenvalue , that is, the following holds:

Also, there are a few differential operators that have deformed the exponential function as eigenfunction.

Let us define the deformed -differential and -derivative accordingly with operation (2.11) (see [15]):

With respect to (2.13), we have The -derivative holds on the property of linearity and the product rule:

Let for or for . For , we define the inverse operator to operator (inverse up to a constant) by

It is easy to prove that

Proposition 3.2. The function is the eigenfunction of the operators and with eigenvalues and , respectively, that is, Moreover, for , the following is valid:

Lemma 3.3. For , , and , the following holds:

Proof. Equalities (3.10) and (3.11) follow from definition (2.7), (3.4), and (3.6). For (3.12), we recall the expansion (2.17) and the formal geometric series:

Furthermore, let us introduce a multiplicative operator

Lemma 3.4. For , the following holds:

Proof. Using the product rule for , we get equalities (3.15) for : Hence, which proves equality (3.16) for . The cases for can be proved by induction or by repeated procedure:

Theorem 3.5. For , the following is valid:

Proof. The statement is obviously true for . Suppose that formula is true for . According to Lemma 3.4, we have

Now, we are able to generalize the special differential operator , stated as the Laguerre derivative in [11, 12], which appears in mathematical modelling of phenomena in viscous fluids and the oscillating chain in mechanics. Substituting the ordinary derivative and variable with the deformed one, we get the deformed Laguerre derivative

Lemma 3.6. For , , and , the following is valid:

Proof. With respect to Proposition 3.2, equality (3.10), and the product rule for , we have wherefrom we get the operational inscription. The second equality follows from the repeated application of (3.10).

At last, we refer to the and operators as the descending (or lowering) and ascending (or raising) operators associated with the polynomial set if

Then, the polynomial set is called quasimonomial with respect to the operators and (see [16]).

It is easy to see that and are the descending and ascending operators associated with the set of generalized monomial . Also, and are the descending and ascending operators associated with the set of generalized monomial .

4. The Functional Sequence Induced by Iterated Deformed Laguerre Derivative

Let , for or for and . We define functions for by the relation

The first members of the functional sequence are

Lemma 4.1. The function is the polynomial of degree in the deformed variable .

Proof. Using equality (3.23), we have With respect to (3.24), repeating the previous step, we get where is a monic polynomial of degree . According to Proposition 2.1, we have

Theorem 4.2. The functions satisfy the next relation of orthogonality: where , and

Proof. Substituting in integral , according to Proposition 2.1, we get Since and , the integral becomes Applying integration by parts twice and using relation (3.10), we get where is a polynomial. Because of we have Repeating the procedure times, we get If , then the following holds: If , then

Notice that the orthogonality relation can be also written in the form

This orthogonality relation and other properties that will be proven indicate that the functions are in close connection with the Laguerre polynomials. That is why we will call them the deformed Laguerre polynomials.

5. Properties of the Deformed Laguerre Polynomials

Let us recall that the Laguerre polynomials defined by [17] satisfy the orthogonality relation the three-term recurrence relations and the differential equations of second order

Theorem 5.1. The functions can be represented by

Proof. Having in mind that , where is a monic polynomial of degree , and changing variable by in integrals for , we have It is a well-known orthogonality relation for the Laguerre polynomials. That is why , where . Since is monic and it must be that and therefore .

The next corollaries express two concepts of orthogonality of these functions.

Corollary 5.2. For , the following is valid: where

From this close connection of functions with the Laguerre polynomials, their properties, as the summation formula, recurrence relation, or differential equation, follow immediately.

Corollary 5.3. The functions have the next hypergeometric representation:

Corollary 5.4. The function is a solution of the differential equation or, in the other form,

Proof. The first form of equation is obtained from the differential equation of the Laguerre polynomials and Theorem 5.1. For the second one, it is enough to notice that

Corollary 5.5. The function is a solution of the differential equation

Theorem 5.6. The sequence has the following generating function:

Proof. Let Notice that According to Theorem 5.1 and recurrence relation (5.3), by summation we get According to (5.17), we have Solving the obtained differential equation with respect to the initial condition , we get

Theorem 5.7. The functions have the following differential properties:

Proof. Applying operator to equality (5.15), according to Proposition 3.2, we get that is, Multiplying by and comparing coefficients, we get equality (5.21). In a similar way, using operator , we get equality (5.23). Equalities (5.22) and (5.24) can be obtained comparing coefficients of powers of , but using expansion

Theorem 5.8. For functions , the next addition formulas are valid:

Proof. We get the first addition formula from Theorem 5.1, equalities (2.11)–(2.14), and the addition formula for the Laguerre polynomials [18]: For the second formula, we consider generating function. According to (5.15) and Proposition 2.1, we have that is, Comparing coefficients of , we get the required equality.

6. The Deformed Laguerre Polynomials in the Context of Operational Calculus

In this section, we consider the deformed Laguerre polynomials from the operational aspect (see [16, 19]). Let us denote

Theorem 6.1. The polynomials have the following operational representations:

Proof. According to Corollary 5.3 and Lemma 3.3, we have In a similar way,

Theorem 6.2. The polynomial set is quasimonomial associated to the descending and ascending operators and , respectively:

Proof. Using the previous theorem, we show that is the descending operator for : Also, is the ascending operator because of

Theorem 6.3. For , the following is valid:

Proof. Using the formal expansion of the exponential function and Lemma 3.6, we have which, according to Corollary 5.3, proves the statement.

Remark 6.4. When and , all properties of functions give corresponding ones for the Laguerre polynomials (see, e.g., [12, 19, 20]).

Acknowledgment

This paper is supported by the Ministry of Sciences and Technology of Republic Serbia, projects no. 174011 and no. 44006.

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