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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 190726, 17 pages
Functions Induced by Iterated Deformed Laguerre Derivative: Analytical and Operational Approach
1Department of Mathematics, Faculty of Electronic Engineering, University of Niš, 18000 Niš, Serbia
2Department of Mathematics, Faculty of Occupational Safety, University of Niš, 18000 Niš, Serbia
3Department 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.
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.
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 , 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 .
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 .
Let us define function by
Since this function can be viewed as a one-parameter deformation of the exponential function of two variables.
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.,  or ). 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 ). 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.
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:
Furthermore, let us introduce a multiplicative operator
Lemma 3.4. For , the following holds:
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 ).
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 .
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  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 : 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
Theorem 6.1. The polynomials have the following operational representations:
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:
This paper is supported by the Ministry of Sciences and Technology of Republic Serbia, projects no. 174011 and no. 44006.
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