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

and Applied Analysis 3 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 eh ( x, y ) exp ( y h ln 1 hx ) . 2.6 Hence, similar to what in 14 , we can use cylinder transformation as deformation function x → {x}h by {x}h 1 h ln 1 hx ln 1 hx 1/h ( x ∈ C \ { − 1 h }) . 2.7 Thus, the following holds: eh ( x, y ) e{x}h . 2.8 We can show that function 2.1 holds on some basic properties of the exponential function. Proposition 2.1. For x ∈ C \ {−1/h} and y, y1, y2 ∈ R, the following holds: eh ( x, y ) > 0 ( x < − 1 h for h < 0 or x > − 1 h for h > 0 ) , eh ( 0, y ) eh x, 0 1, e−h ( x, y ) eh −x,−y ( x / 1 h ) , eh ( x, y1 y2 ) eh ( x, y1 ) eh ( x, y2 ) . 2.9 Notice that the additional property is true with respect to the second variable only. However, with respect to the first variable, the following holds: eh ( x1, y ) eh ( x2, y ) eh ( x1 x2 hx1x2, y ) . 2.10 This equality suggests introducing a generalization of the sum operation x1⊕hx2 x1 x2 hx1x2. 2.11 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 x / − 1/h, the h-inverse exists as hx −x 1 hx , 2.12 4 Abstract and Applied Analysis and x⊕h hx 0 is valid. Hence, I,⊕h is an abelian group, where I −∞,−1/h for h < 0 or I −1/h, ∞ for h > 0. In this way, the h-subtraction can be defined by x1 hx2 x1⊕h hx2 x1 − x2 1 hx2 ( x2 / − 1 h ) . 2.13 With respect to 2.7 , we can prove the next equality for x1, x2 ∈ I: {x1}h {x2}h {x1⊕hx2}h. 2.14 Proposition 2.2. For x1, x2 ∈ C \ {−1/h} and y ∈ R, the following is valid: eh ( x1⊕hx2, y ) eh ( x1, y ) eh ( x2, y ) , eh ( x1 hx2, y ) eh ( x1, y ) eh ( x2,−y ) . 2.15 In order to find the expansions of the introduced deformed exponential function, we introduce the generalized backward integer power given by z 0,h 1, z n,h n−1 ∏ k 0 z − kh n ∈ N, h ∈ R \ {0} . 2.16 Proposition 2.3. For function x, y → eh x, y , the following representation holds: eh ( x, y ) ∞ ∑ n 0 {x}hy n! , eh ( x, y ) ∞ ∑ n 0 xy n,h n! |hx| < 1 . 2.17 Remark 2.4. Notice that in expressions 2.8 and the first expansion in 2.17 the deformation of variable x appears, but, contrary to that in the second expansion in 2.17 , the deformation of powers of y is present. 3. The Deformed Operators Let us recall that the h-difference operator is Δz,hf z f z h − f z h . 3.1 Proposition 3.1 see 8 . The function y → eh x, y is the eigenfunction of difference operatorΔy,h with eigenvalue x, that is, the following holds: Δy,h eh ( x, y ) x eh ( x, y ) . 3.2 Abstract and Applied Analysis 5 Also, there are a few differential operators that have deformed the exponential function as eigenfunction. Let us define the deformed h-differential and h-derivative accordingly with operation 2.11 see 15 :and Applied Analysis 5 Also, there are a few differential operators that have deformed the exponential function as eigenfunction. Let us define the deformed h-differential and h-derivative accordingly with operation 2.11 see 15 : dhz lim u→ z z hu, Dz,hf z df z dhz lim u→ z f z − f u z hu . 3.3 With respect to 2.13 , we have Dz,hf z df z dhz lim u→ z f z − f u z − u / 1 hu 1 hz df z dz . 3.4 The h-derivative holds on the property of linearity and the product rule: Dz,h ( αf z βg z ) αDz,hf z βDz,hg z , Dz,h ( f z g z ) f z Dz,hg z g z Dz,hf z . 3.5 Let I −∞,−1/h for h < 0 or I −1/h, ∞ for h > 0. For x ∈ I, we define the inverse operator to operator Dx,h inverse up to a constant by D−1 x,hf x ∫x


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 .

The Deformed Exponential Functions
In this section we will present a deformation of an exponential function of two variables depending on parameter h ∈ R \ {0}, which is introduced in 8 .Let us define function x, y → e h x, y by e h x, y 1 hx y/h x ∈ C \ − 1 h , y ∈ R .

2.1
Since lim h → 0 e h x, y e xy , 2.2 this function can be viewed as a one-parameter deformation of the exponential function of two variables.If h 1 − q q / 1 and y 1, function 2.1 becomes that is, e 1−q x, 1 e x q , where e x q is Tsallis q-exponential function 1 defined by

2.4
If h p − 1 p / 1 and x 1, function 2.1 becomes Thus, the following holds: e h x, y e {x} h y .

2.8
We can show that function 2.1 holds on some basic properties of the exponential function.

2.9
Notice that the additional property is true with respect to the second variable only.However, with respect to the first variable, the following holds:

2.10
This equality suggests introducing a generalization of the sum operation

2.11
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 x / − 1/h, the h -inverse exists as 12 and x⊕ h h x 0 is valid.Hence, I, ⊕ h is an abelian group, where I −∞, −1/h for h < 0 or I −1/h, ∞ for h > 0. In this way, the h -subtraction can be defined by With respect to 2.7 , we can prove the next equality for x 1 , x 2 ∈ I:

2.15
In order to find the expansions of the introduced deformed exponential function, we introduce the generalized backward integer power given by x n y n,h n! |hx| < 1 .

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

The Deformed Operators
Let us recall that the h-difference operator is With respect to 2.13 , we have

3.4
The h-derivative holds on the property of linearity and the product rule: For x ∈ I, we define the inverse operator to operator D x,h inverse up to a constant by It is easy to prove that The function e h x, y is the eigenfunction of the operators D x,h and ∂/∂y with eigenvalues y and {x} h , respectively, that is,

D x,h e h x, y ye h x, y , ∂ ∂y e h x, y
{x} h e h x, y .

3.13
Furthermore, let us introduce a multiplicative operator For n ∈ N 0 , the following holds:

3.16
Proof.Using the product rule for D x,h , we get equalities 3.15 for n 1: Hence, which proves equality 3.16 for n 1.The cases for n > 1 can be proved by induction or by repeated procedure:

3.19
Theorem 3.5.For n ∈ N 0 , the following is valid: Proof.The statement is obviously true for n 1. Suppose that formula is true for n.According to Lemma 3.4, we have x,h 0.

3.21
Now, we are able to generalize the special differential operator d/dx x d/dx , 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

3.22
Lemma 3.6.For x ∈ I, y ∈ R, and k, n ∈ N 0 , the following is valid: Proof.With respect to Proposition 3.2, equality 3.10 , and the product rule for D x,h , we have

D x,h X h D x,h e h x, y D x,h {x} h ye h x, y y 1 y{x} h e h x, y ,
3.25 wherefrom we get the operational inscription.The second equality follows from the repeated application of 3.10 .
At last, we refer to the M and P operators as the descending or lowering and ascending or raising operators associated with the polynomial set {q n } n∈N 0 if M q n nq n−1 , P q n q n 1 .

3.26
Then, the polynomial set {q n } n∈N 0 is called quasimonomial with respect to the operators M and P see 16 .
It is easy to see that D x,h and X h are the descending and ascending operators associated with the set of generalized monomial {x} n h n ∈ N 0 .Also, D x,h X h D x,h and D −1 x,h are the descending and ascending operators associated with the set of generalized monomial {x} n h /n! n ∈ N 0 .

The Functional Sequence Induced by Iterated Deformed Laguerre Derivative
Let h / 0, I −∞, −1/h for h < 0 or I −1/h, ∞ for h > 0 and G I × R .We define functions x, y → L n,h x, y for x, y ∈ G n ∈ N 0 by the relation The first members of the functional sequence {L n,h x, y } n∈N 0 are where k, n ∈ N 0 , k ≤ n, and

4.7
Abstract and Applied Analysis 9 Proof.Substituting L n,h x, y in integral J n,k , according to Proposition 2.1, we get 4.9 and e h x, −h 1 hx −1 , the integral becomes Applying integration by parts twice and using relation 3.10 , we get where q is a polynomial.Because of Repeating the procedure k times, we get

4.14
If n > k, then the following holds:

4.15
If n k, then

4.16
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 L n,h x, y are in close connection with the Laguerre polynomials.That is why we will call them the deformed Laguerre polynomials.

Properties of the Deformed Laguerre Polynomials
Let us recall that the Laguerre polynomials defined by 17 the three-term recurrence relations and the differential equations of second order Theorem 5.1.The functions L n,h x, y n ∈ N 0 can be represented by L n,h x, y L n y{x} h .

5.5
Proof.Having in mind that L n,h x, y −1 n Q n y{x} h , where Q n is a monic polynomial of degree n, and changing variable x by t y{x} h in integrals J n,k for k ≤ n, we have It is a well-known orthogonality relation for the Laguerre polynomials.That is why Q n t c n L n t , where c n const.Since Q n is monic and it must be that c n −1 n and therefore L n,h x, y L n t L n y{x} h .
The next corollaries express two concepts of orthogonality of these functions.
Corollary 5.2.For m, n ∈ N 0 , the following is valid: where

5.9
From this close connection of functions L n,h x, y with the Laguerre polynomials, their properties, as the summation formula, recurrence relation, or differential equation, follow immediately.
Corollary 5.3.The functions L n,h x, y n ∈ N 0 have the next hypergeometric representation: Corollary 5.4.The function x → L n,h x, y y > 0 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

5.13
Corollary 5.5.The function y → L n,h x, y x ∈ I is a solution of the differential equation Theorem 5.6.The sequence {L n,h x, y } n∈N 0 has the following generating function:

5.18
Abstract and Applied Analysis 13 According to 5.17 , we have

5.26
Multiplying by 1−t and comparing coefficients, we get equality 5.21 .In a similar way, using operator ∂/∂y, we get equality 5.23 .Equalities 5.22 and 5.24 can be obtained comparing coefficients of powers of t, but using expansion

5.27
Theorem 5.8.For functions {L n,h x, y }, the next addition formulas are valid:

5.28
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 :

5.29
For the second formula, we consider generating function.According to 5.15 and Proposition 2.1, we have

5.31
Comparing coefficients of t n , we get the required equality.

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 x,h n 1 .

6.3
In a similar way, x, y .

6.4
Theorem 6.2.The polynomial set { L n,h x, y } n∈N 0 is quasimonomial associated to the descending and ascending operators M x and P x , respectively: x,h , P x − 1 y D x,h X h D x,h .6.5 Proof.Using the previous theorem, we show that M x is the descending operator for { L n,h x, y }: L n 1,h x, y .6.6 Also, P x is the ascending operator because of
4.2 Lemma 4.1.The function L n,h x, y n ∈ N 0 is the polynomial of degree n in the deformed variable y{x} h ln 1 hx y/h ln e h x, y .Proof.Using equality 3.23 , we have D x,h X h D x,h e h x, −y y y{x} h − 1 e h x, −y .n is a monic polynomial of degree n.According to Proposition 2.1, we have L n,h x, y −1 n Q n y{x} h .4.5 Theorem 4.2.The functions L n,h x, y satisfy the next relation of orthogonality: Proof.According to Corollary 5.3 and Lemma 3.3, we have h x, y .Theorem 6.3.For L n,h x, y , the following is valid: Proof.Using the formal expansion of the exponential function and Lemma 3.6, we have Remark 6.4.When h → 0 and y 1, all properties of functions L n,h x, y give corresponding ones for the Laguerre polynomials see, e.g.,12, 19, 20 .