# The Extended Laguerre PolynomialsAq,nαx Involving Fqq,q>2

Authors: Adnan Khan; Muhammad Kalim; Ali Akgül; Fahd Jarad
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, for the proposed extended Laguerre polynomialsAαq,nx, the generalized hypergeometric function of the type Fqq,q>2 and extension of the Laguerre polynomial are introduced. Similar to those related to the Laguerre polynomials, the generating function, recurrence relations, and Rodrigue’s formula are determined. Some corollaries are also discussed at the end.


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## Body

## 1. Introduction and Applications

Due to its wide applications, the study of orthogonal polynomials has been a popular research topic for many years. Many of these polynomials are generated by hypergeometric functions. Indeed, the orthogonal polynomials have numerous properties of interest, e.g., recurrence relations and differential equations. Based on their Rodrigues formulae, generating functions and solutions of integral equations with orthogonal polynomials as kernels have been extensively investigated.Generalizations and extensions of orthogonal polynomials are in the another familiar direction of research. One of the polynomial set which has been extended is a set of Laguerre polynomials. Laguerre polynomials are well-known to form an orthogonal set with respect to the weight functionzαe−z on the interval 0,∞.A set of Laguerre polynomials is generated by well-known confluent hypergeometric function1F1. It can be also generated by hypergeometric function 0F1. Another direction is the study of Laguerre polynomials based on more than one variable which are often used in physical and statistical model. One, too, combinatorial polynomial images, moments, orthogonality relation, and a combinatorial understanding Ikyrana coefficients Al-Salam and Chihara q Laguerre polynomial, can study various aspects. Orthogonal polynomials, namely, Hermite polynomials and Legendre polynomials can also be studied through the finite series involving Laguerre polynomials.Laguerre polynomials are used to solve noncentral Chi-square distribution. Laguerre polynomials are the orthogonal polynomial satisfied the recurrence relations. Various specializations are studied with application to classical orthogonal polynomials. Kinetic theory of particles based on Laguerre polynomial macroscopic hydrodynamic quantities and kinetic coefficients of different medium is used to set.There are a large number of generalizations and extensions of Laguerre polynomials, e.g., Shively’s polynomials. Many of these generalizations are based on its Rodrigues formulae in addition to hypergeometric functions. Recently, an interesting integral representation of generalized hypergeometric functions has been determined. It is now natural to point to a generalization of Laguerre polynomials based on such a discovery. This idea has motivated the current work. Also, it will explore deeper investigation and extensions of results which we proved in our early studies and research.In this work, we discuss the features of Extending Laguerre polynomial involvingFqq,q>2. Extending Laguerre polynomial set has been a popular research issue well considered for years. There have a number of directions to do so. One direction is to follow the definition of Laguerre polynomials based on the confluent hypergeometric function, explicitly
(1)Lαnx=1+αnn!F11−n;1+α;x.Shively [1] extended the Laguerre polynomials as
(2)Rna,x=a2nn!anF11−n;a+n;x.He used a factora+n instead of 1+α in Laguerre polynomials. In his study, he found a large number of its properties including the result that a finite sum of Laguerre polynomials is Shively’s polynomials. Habibullah [2] proved the Rodrigues formula for Shively’s polynomials in the following form
(3)Rna+1,x=exx−α−nn!Dnxα+2ne−x,similar to the Rodrigues formula
(4)Lnαx=exx−αn!Dnxα+ne−x,for the Laguerre polynomials.Researchers have also often based their generalization on extension of Rodrigues formula and subsequently determined properties of extended polynomials. Chatterjea [3] developed an extension of the Laguerre polynomial by strengthening the Rodrigues formula. Chatterjea and Das [4] restructured their definition and the resultant study by considering another version of the Laguerre polynomials.Chen and Srivastava [5] found a stronger Rodrigues formula to develop a generalization of the Laguerre polynomial.The forms generalized Rodrigues formulae by Chak [6] show that robust following of this method of defining extensions of the Laguerre polynomial. Since comprehensive literature is available on special functions, we follow Shively’s tradition to introduce the definition of the extended Laguerre polynomials set based on special functions similar to that contained the original definition.Dattoli et al. [7] used an exponential generating functions approach involving Hermite polynomials and Bessel functions introduced new families. He, too, studied their respective recurrence relations and showed that they fulfill different differential equations. Trickovic and Stankovic [8] of the Jacobi and Laguerre polynomial orthogonality of rational functions that have proved equally. Trickovic and Stankovic [8] have proved the orthogonality of the Jacobi and the Laguerre polynomials.Khan and Shukla [9] have introduced a novel method to give operator representations of certain polynomials. They gave binomial and trinomial operators representations of certain polynomials. Grinshpan [10] has shown that all solutions to the equations of a family of integral equations fulfill modulus inequality. Duenas et al. [11] a derivative of a Dirac delta by adding a perturbation of a Laguerre-Hahn functional gain catalog.Kim et al. [12] have studied some interesting identities and also studied Bernoulli and Euler’s numbers in connection with the properties of Laguerre polynomials. They derived identities by using the orthogonality of Laguerre polynomials w.r.t the relevant inner product. Marinkovic et al. [13] have demonstrated the theory of deformed Laguerre derivative defined by iterated deformed Laguerre operator. Nowak et al. [14] convolution type Laguerre function expansions in order to prove the standard estimates has developed a technique. Khan and Habibullah [15] have introduced A2,nx=F22−n/2,−n+1/2;1/2,1;x2.Khan and Kalim [16] have introduced
(5)A3,mαy=1+αmm!F33−m3,−m+13,−m+23;1+α3,2+α3,3+α3;y3.Doha et al. [17] modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients, and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced.Dattoli et al. [18] applied operational techniques to introduce suitable families of special functions. Andrews et al. [19], Trickovic and Stankovic [20], Radulescu [21], and Doha and Youssri [22] have done a lot of work for properties of Laguerre polynomials. Akbary et al. [23] can be referred for other applications of Laguerre polynomials. Li [24], Aksoy et al. [25], Wang [26], and Krasikov and Zarkh [27] have studied problems of permutation of polynomials, bijections that can induce polynomials with integer coefficients is modulo m.We organize our manuscript as: we present the properties and applications of extended polynomials in Section2. We give the extended Laguerre polynomials in Section 3. We discuss the generating functions in Section 4. We present the recurrence relations in Section 5. We give the differential equations in Section 6. We discuss the Rodrigues formula in Section 7. We give the special properties in Section 8. We present some other generating functions in Section 9. We give the expansion of the polynomials in Section 10. We present the conclusion in the last section.


## 2. Extended Polynomial Properties and Application Elementary Results

Das [28] has modified the work of Al-Salam [29]. Carlitz [30] has given a generating function and an explicit polynomial expression for the polynomial Yncx;k, a variant of Laguerre polynomials. Srivastava [31] has derived the several bilinear generating functions by using generalized hypergoemetric functions. Explicitly, we can mention [Erdélyi p, 190] [32].
(6)Dmxα+mLnα+mx=Γα+m+n+1Γα+n+1xαLnαx,D=ddx.One generalization of Laguerre polynomials isRna,x as Shively defined it by [Rainville p, 298] [33].
(7)Rna,x=a2nn!anF11−n;a+n;x,he used these results as an integral equation involving Shively’s polynomials. Karande and Thakare [34] have derived the generating functions, bilinear generating functions, and recurrence relations by using the biorthogonal set of Konhausar. Panda [35] has studied a new generalization based on several known polynomials systems belonging to the families of the classical Jacobi, Hermite, and Laguerre polynomials. Parashar [36] has introduced a new set of Laguerre polynomials Lnα,hx related to the Laguerre polynomials Lnαx. Sharma and Chongdar [37] have proved an extension of bilateral generating functions of the modified Laguerre polynomials.Lemma 1.
Ifj∈ℤ+ and n is any nonnegative integer, then
(8)−nqj−n+1qj⋯−n+q−1qj=−1qjn!qqjn−qj!.Proof.
(9)−nqj−n+1qj⋯−n+q−1qj=−nq−nq+1−nq+2⋯−nq+j−1,−n+1q−n+1q+1−n+1q+2⋯−n+1q+j−1,..⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯−n+q−1q−n+q−1q+1−n+q−1q+2⋯−n+q−1q+j−1=−nq−n+qq−n+2qq⋯−n+qj−qq,−n+1q−n+q+1q−n+2q+1q⋯−n+qj−q+1q,..⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯−n+q−1q−n+2q−1q−n+3q−1q⋯−n+qj−1q=−1qjn!qjqjn−qj!.Lemma 2.
Ifk∈ℤ+ and n is any nonnegative integer, thus
(10)αkn=kknαknα+1kn⋯α+k−1kn,
Rainville [33] (p 22).Lemma 3.
Ifk∈ℤ+ and n is any nonnegative integer, thus
(11)∑n=0∞∑k=0nBk,n=∑n=0∞∑k=0∞Bk,n+k,
Rainville [33] (p 57).Lemma 4.
Ifk∈ℤ+ and n is any nonnegative integer, then
(12)∑n=0∞∑k=0∞Bk,n=∑n=0∞∑k=0nBk,n−k,
Rainville [33] (p 56).


## 3. The Extended Laguerre PolynomialsAαq,nx

We define the extended Laguerre polynomial setAαq,nx by
(13)Aαq,nx=exq+αnn!Fqq−nq,−n+1q,⋯,−n+q−1q;;xqq+αq,q+1+αq,⋯,2q+α−1q,where α∈ℝ, n,q∈ℤ+.Theorem 5.
IfAαq,nx are the extended Laguerre polynomials, then
(14)Aαq,nx=exq+αn∑j=0nq−1qjn−qj!q+αqjxqjqj!,α∈ℝ,n∈ℤ+.Proof.
Consider(15)Aαq,nx=exq+αnn!Fqq−nq,−n+1q,⋯,−n+q−1q;;xqq+αq,q+1+αq,⋯,2q+α−1q=exq+αnn!×∑j=0nq−n/qj−n+1/qj⋯−n+q−1/qjq+α/qjq+1+α/qj⋯2q+α−1/qjxqjqj!.
By using Lemma1(16)Aαq,nx=exq+αnn!×∑j=0nq−1qjn!qqjn−qj!q+α/qjq+1+α/qj⋯2q+α−1/qjxqjqj!.
Then from Lemma2, we have
(17)Aαq,nx=exq+αn∑j=0nq−1qjn−qj!q+αqjxqjqj!.


## 4. Generating Functions

The following theorem formulates a generating function for the extended Laguerre polynomialsAαq,nx.Theorem 6.
Ifn,j∈ℤ+, then
(18)∑n=0∞∑j=0nq−1qjextnn−qj!q+αqjxqjqj!=ex+tF0q−−;q+αq,q+1+αq,⋯,2q+α−1q;−xtqq.Proof.
By using Lemma3, we acquire
(19)∑n=0∞∑j=0nq−1qjextnn−qj!q+αqjxqjqj!=∑n=0∞∑j=0∞−1qjextn+qjn!q+αqjxqjqj!=ex∑n=0∞tnn!∑j=0∞−1qktqjq+αqjxqjqj!=ex+t∑j=0∞−xtqjq+αqjqj!.
By using Lemma2, we acquire
(20)∑n=0∞∑j=0nq−1qjextnn−qj!q+αqjxqjqj!=ex+t∑j=0∞−xtqjqqjq+α/qjq+1+α/qj⋯2q+α−1/qjqj!=ex+tF0q−−;q+αq,q+1+αq,⋯,2q+α−1q;−xtqq.Corollary 7.
Ifα∈ℝ and n,q,j∈ℤ+, then
(21)∑n=0∞Aαq,nxtnq+αn=ex+tF0q−−;−xtqqq+αq,q+1+αq,⋯,2q+α−1q;.Proof.
From Equation (14), we acquire
(22)∑n=0∞Aαq,nxq+αntn=∑n=0∞∑j=0nq−1qjn−qj!q+αqjxqjqj!tn.
A use of Theorem (18), therefore, shows that the extended Laguerre polynomials have the generating function given by
(23)∑n=0∞Aαq,nxtnq+αn=ex+tF0q−−;−xtqqq+αq,q+1+αq,⋯,2q+α−1q;.Theorem 8.
Ifc∈ℤ+, then
(24)∑n=0∞cnAαq,nxtnq+αn=ex1−tcFqqcq,c+1q,⋯,c+q−1q;−xt1−tqq+αq,q+1+αq,⋯,2q+α−1q;.Proof.
From Equation (22), we note that
(25)∑n=0∞cnAαq,nxq+αntn=∑n=0∞cnex∑j=0nq−1qjn−qj!q+αqjxqjqj!tn.
By using Lemma3, we acquire
(26)∑n=0∞cnAαq,nxtnq+αn=∑n=0∞∑j=0∞cn+qjextn+qjn!−1qjxqjq+αqjqj!=∑j=0∞∑n=0∞c+qjntnn!cqjq+αqjex−xtqjqj!.
Sincecn+qj=c+qjncqj and 1−t−m=∑n=0∞mntn/n!, it thus implies that
(27)∑n=0∞cnAαq,nxtnq+αn=∑j=0∞cqj1−tc+qjq+αqjex−xtqjqj!=ex1−tc∑k=0∞cqjq+αqj1qj!−xt1−tqj.
By using Lemma2, we consequently obtain the required result
(28)∑n=0∞cnAαq,nxtnq+αn=ex1−tcFqqcq,c+1q,⋯,c+q−1q;−xt1−tqq+αq,q+1+αq,⋯,2q+α−1q;.Corollary 9.
Ifα∈ℝ and n,m,j∈ℤ+, then
(29)∑n=0∞Aαq,nxtn=11−tq+αexpx−2xt1−t.Proof.
Putc=q+α in Equation (24), we obtain our desired result.


## 5. Recurrence Relations

We describe the recurrence relations for the extended Laguerre polynomialsAαq,nx.Theorem 10.
Ifα∈ℝ and n,j∈ℤ+, then
(30)xDAαq,nx=n+xAαq,nx−q+α+n−1Aαq,n−1x,D=ddx.Proof.
From Equation (18)
(31)∑n=0∞Aαq,nxtnq+αn=ex+tF0q−−;−xtqqq+αq,q+1+αq,⋯,2q+α−1q;.
Letσq,nx=Aαq,nx/q+αn.
Suppose that(32)F0q−−;−xtqqq+αq,q+1+αq,⋯,2q+α−1q;=ψxqtqq.(33)ThenF=ex+tψxqtqq=∑n=0∞σq,nxtn,
provide that the series is uniformly convergent. By taking partial derivatives,(34)∂F∂x=ex+tψ+xq−1tqex+tψ′,(35)∂F∂t=ex+tψ+xqtq−1ex+tψ′,(36)x∂F∂x−t∂F∂t=xF−tF.
Now, sinceF=∑n=0∞σq,nxtn, therefore,
(37)∂F∂x=∑n=0∞σq,n′xtnandt∂F∂t=∑n=0∞nσq,nxtn.
Equation (36) then yields
(38)x∑n=0∞σq,n′xtn−∑n=0∞nσq,nxtn=x∑n=0∞σq,nxtn−∑n=0∞σq,nxtn+1=x∑n=0∞σq,nxtn−∑n=1∞σq,n−1xtn.
We getσ2,0′x=0, and for n>1,
(39)xσq,n′x−nσq,nx=xσq,nx−σq,n−1x.
This implies that(40)xDAαq,nx=n+xAαq,nx−q+α+n−1Aαq,n−1x.Theorem 11.
Ifα∈ℝ and n≥2 then
(41)DAαq,nx=DAαq,n−1x+Aαq,nx−2Aαq,n−1x.Proof.
From Equation (29), we get the following
(42)1−t−q−αexpx1−2t1−t=∑n=0∞Aαq,nxtn.(43)LetF=Atexpx1−2t1−t=∑n=0∞yq,nxtn,(44)∂F∂x=1−2t1−tAtexpx1−2t1−t,(45)1−t∂F∂x=1−2tAtexpx1−2t1−t.
By using Equation (42), we obtain
(46)1−t∂F∂x=1−2tF.(47)SinceF=∑n=0∞yq,nxtn,thereforewehave∂F∂x=∑n=0∞yq,n′xtn.
Equation (46) can be expressed as
(48)∑n=0∞yq,n′xtn−∑n=0∞yq,n′xtn+1=∑n=0∞yq,nxtn−2∑n=0∞yq,nxtn+1,(49)∑n=0∞yq,n′xtn−∑n=1∞yq,n−1′xtn=∑n=0∞yq,nxtn−2∑n=1∞yq,n−1xtn.
We reachyq,0′x=0,yq,1′x=0 and for n>2,
(50)DAαq,nx=DAαq,n−1x+Aαq,nx−2Aαq,n−1x.Theorem 12.
Ifα∈ℝ and n≥q, then
(51)DAαq,nx=Aαq,nx−∑j=0n−1Aαq,jx.Proof.
Equation (46) can be written as
(52)∂F∂x=1−t1−tF.
By using Equation (42), we obtain
(53)∂F∂x=1−t1−t∑n=0∞yq,nxtn.
By using Equation (47), we obtain.
(54)∑n=0∞yq,n′xtn=∑n=0∞yq,nxtn−∑n=0∞tn+1∑n=0∞yq,nxtn=∑n=0∞yq,nxtn−∑n=0∞∑j=0∞yq,jxtjtn+1.
Since∑n=0∞∑k=0∞Bk,n=∑n=0∞∑k=0nBk,n−k, (Rainville [33], (p 56)).
(55)∑n=0∞yq,n′xtn=∑n=0∞yq,nxtn−∑n=0∞∑j=0nyq,jxtn+1=∑n=0∞yq,nxtn−∑n=1∞∑j=0n−1yq,jxtn.
It follows thatyq,0′x=0,yq,1′x=0 and for n>q,yq,n′x=yq,nx−∑j=0n−1yq,jx, and DAαq,nx=Aαq,nx−∑j=0n−1Aαq,jx.Theorem 13.
Ifα∈ℝ and n≥q+1, then
(56)nAαq,nx=3x−q−αAαq,n−1x−q+α+n−2Aαq,n−2x.Proof.
We can have the following equation after eliminating the derivatives from Equations (30) and (41).
(57)0=nAαq,nx−xDAαq,n−1x+2x−q−α−n+1Aαq,n−1xnAαq,nx=xDAαq,n−1x−2x−q−α−n+1Aαq,n−1x.
Now, by using Equation (30), we finally have
(58)nAαq,nx=n−1+xAαq,n−1x−q+α+n−2Aαq,n−2x+2x−q−α−n+1Aαq,n−1x,(59)nAαq,nx=3x−q−αAαq,n−1x−q+α+n−2Aαq,n−2x.Theorem 14.
Ifα∈ℝ and n,q,j∈ℤ+, then
(60)A1+αq,n−1x+Aαq,nx=A1+αq,nx..Proof.
From Equation (14), we obtain
(61)A1+αq,n−1x=exq+1+αn−1∑j=0n−1q−1qjn−1−qj!q+1+αqjxqjqj!,so that Aαq,nx=exq+αn∑j=0n/q−1qj/n−qj!q+αqjxqj/qj!.
By adding the above equations, we get(62)Aq,n−11+ax+Aq,nax=exq+1+αn−1∑j=0n−1q−1qjn−1−qj!q+1+αqjxqjqj!+exq+αn∑j=0nq−1qjn−qj!q+αqjxqjqj!=ex∑j=0n−1qq+α+n−1!−1qjn−1−qj!q+α+qj!xqjqj!+∑k=0nqq+α+n−1!−1qjn−qj!q+α+qj−1!xqjqj!=ex∑k=0n−1qq+α+n−1!−1qjn−1−qj!q+α+qj!xqjqj!+∑j=0n−1qq+α+n−1!−1qjn−qj!q+α+qj−1!xqjqj!+xqnqn!=ex∑k=0n−1qq+α+n−1!xqj−1qjqj!1n−1−qj!q+α+qj!+1n−qj!q+α+qj−1!+xqnqn!=ex∑k=0n−1qq+α+n−1!−1qjn−qj!q+α+qj!q+α+nxqjqj!+xqnqn!=ex∑j=0n−1qq+α+n!−1qjn−qj!q+α+qj!xqjqj!+xqnqn!=exq+1+αn∑j=0nq−1qjn−qj!q+1+αqjxqjqj!=A1+αq,nx.


## 6. Differential Equation

Since the Extended Laguerre polynomial is a constant multiple of hypergeometric functionsFqq, we may obtain the differential equation.Theorem 15.
Ifα∈ℝ and n≥q, then
(63)xD2Aαq,nx+q+α−3xDAαq,nx+2x+n−q−αAαq,nx=0.Proof.
By taking partial derivatives of Equation (30), we have
(64)xD2Aq2,nαx+DAq,nαx=n+xDAq,nαx+Aq,nαx−q+α+n−1DAq,n−1αx.
By using Equation (41), we have
(65)xD2Aαq,nx+DAαq,nx=n+xDAαq,nx+Aαq,nx−q+α+n−1DAαq,nx−Aαq,nx+2Aαq,n−1x,or
(66)xD2Aαq,nx+q+α−xDAαq,nx=q+α+nAαq,nx−2q+α+n−1Aαq,n−1x.
By using Equation (30), we have
(67)xD2Aαq,nx+q+α−xDAαq,nx=q+α+nAαq,nx+2xDAαq,nx−2n+xDAαq,nx,or
(68)xD2Aαq,nx+q+α−3xDAαq,nx+2x+n−q−αAαq,nx=0.


## 7. Rodrigues Formula

The Rodrigues formula for the Laguerre polynomials is presented as(69)Lαnx=x−αexn!Dnxα+ne−x,but we intend to extend this Rodrigues formula.Theorem 16.
Ifα∈ℝ and n,j∈ℤ+, then
(70)Aαq,nx=x−q−1−αe2xn!Dnxq−1+α+ne−x.Proof.
Consider the extended Laguerre polynomials involvingFqq,q>2(71)Aαq,nx=exq+αnn!Fqq−nq,−n+1q,⋯,−n+q−1q;xqq+αq,q+1+αq,⋯,2q+α−1q;.
By Theorem (14), we have
(72)Aαq,nx=exn!∑j=0nqn!n−qj!qj!q+αnxqjq+αqj=exx−q−1−αn!∑j=0nq−1qjn!n−qj!qj!q+mnxqj+α+q−1q+mqj.
SinceDn−qjxn+α+q−1=q+αnxqj+α+q−1/q+αqj, therefore, we write it as
(73)Aαq,nx=x−q−1−αe2xn!∑j=0nqn!n−qj!qj!−1qje−xDn−qjxn+α+q−1=x−q−1−αe2xn!∑j=0nqnCqjDn−qjxn+α+q−1Dqje−x.
Lastly, we use the Leibnitz formula for thenth derivative to obtain the following
(74)Aαq,nx=x−q−1−αe2xn!Dnxq−1+α+ne−x.


## 8. Special Properties

In this section, we determine the special features of the extended Laguerre polynomialsAαq,nx.Theorem 17.
Ifα,β∈ℝ and n,j,q∈ℤ+, then
(75)Aαq,nx=∑j=0nqα−βqjAβq,n−qjxqj!.Proof.
From Equation (29)
(76)∑n=0∞Aαq,nxtn=11−tq+αexpx1−2t1−t.
Also, consider(77)11−tq+αexpx1−2t1−t=1−t−α−β1−t−q−βexpx1−2t1−t,(78)∑n=0∞Aαq,nxtn=1−t−α−β∑n=0∞Aβq,nxtn=∑n=0∞α−βqntqnqn!∑n=0∞Aβq,nxtn=∑n=0∞∑j=0∞α−βqjtqjAβq,nxtnqj!.
By utilizing Lemma4, we acquire
(79)∑n=0∞Aαq,nxtn=∑n=0∞∑j=0nqα−βqjtqjAβq,n−qjxtn−qjqj!=∑n=0∞∑j=0nqα−βqjAβq,n−qjxtnqj!.
On comparing the coefficients oftn, we acquire
(80)Aαq,nx=∑j=0nqα−βqjAβq,n−qjxqj!.Theorem 18.
Ifα∈ℝ and n,j∈ℤ+, then
(81)Aα+β+qq,nx+y=∑j=0nqAβq,n−qjyAαq,qjx.Proof.
Consider(82)1−t−q−αexpx1−2t1−t1−t−q−βexpy1−2t1−t=1−t−q−α+β+qexpx+y1−2t1−t.
By using Equation (75), we acquire
(83)∑n=0∞Aαq,nxtn∑n=0∞Aβq,nytn=∑n=0∞Aα+β+qq,nx+ytn,(84)∑n=0∞Aα+β+qq,nx+ytn=∑n=0∞∑j=0∞Aβq,nytnAαq,qjxtqj.
By using Lemma4, we acquire
(85)∑n=0∞Aα+β+qq,nx+ytn=∑n=0∞∑j=0nqAβq,n−qjyAαq,qjxtn.
On comparing the coefficients oftn, we acquire
(86)Aα+β+qq,nx+y=∑j=0nqAβq,n−qjyAαq,qjx.Theorem 19.
Ifα∈ℝ and n,j∈ℤ+, then
(87)Aαq,nxy=∑j=0nqq+αnAαq,qjxqyqjq+αqj1−yn−qjn−qj!.Proof.
Consider(88)ex+tF0q−−;−xytqqq+αq,q+1+αq,⋯,2q+α−1q;=e1−ytex+ytF0q−−;−xytqqq+αq,q+1+αq,⋯,2q+α−1q;.
By using Equation (21), we get
(89)∑n=0∞Aαq,nxytnq+αn=∑n=0∞1−yntnn!∑n=0∞Aαq,nxyntnq+αn=∑n=0∞∑j=0∞Aαq,qjxyqjtqjq+αqj1−yntnn!.
By using Lemma4, we acquire
(90)∑n=0∞Aαq,nxytnq+αn=∑n=0∞∑j=0nqAαq,qjxyqjtqjq+αqj1−yn−qjtn−qjn−qj!=∑n=0∞∑j=0nqAαq,qjxyqjq+αqj1−yn−qjtnn−qj!.
On comparing the coefficients oftn, we get
(91)Aαq,nxy=∑j=0nqq+αnAαq,qjxqyqjq+αqj1−yn−qjn−qj!.Theorem 20.
Ifα∈ℝ and n,j,q∈ℤ+, then
(92)∑n=0∞n+qj!Aαq,n+qjxtnqj!n!=1−t−q−α−qjexp−xt1−tAαq,jx1−t.Proof.
Consider the series(93)∑j=0∞∑n=0∞n+qj!Aαq,n+qjxtnyqjqj!n!=∑n=0∞∑j=0nqn!Aαq,nxtn−qjyqjqj!n−qj!=∑n=0∞Aαq,nx∑j=0nqnCqjtn−qjyqj=∑n=0∞Aαq,nxt+yn=1−t−y−q−αexpx1−2y−2t1−y−t.
Since1−t−y−q−α=1−t−q−α1−y/1−t−q−α(94)expx1−2y−2t1−t=expxexp−xy+t1−t−y=expxexp−xt1−texp−x/1−ty/1−t1−y/1−t.
Therefore, Equation (93) becomes
(95)∑j=0∞∑n=0∞n+qj!Aαq,n+qjxqtnyqjqj!n!=1−t−q−α1−y1−t−q−αexpxexp−xt1−texp−x/1−ty/1−t1−y/1−t.
By using Equation (29), we get
(96)∑j=0∞∑n=0∞n+qj!Aαq,n+qjxqtnyqjqj!n!=1−t−q−αexp−xt1−t∑j=0∞Aαq,jx1−ty1−tqj.
On comparing the coefficients ofyqj, we get
(97)∑n=0∞n+qj!Aαq,n+qjxqtnqj!n!=1−t−q−α−qjexp−xt1−tAαq,jx1−t.


## 9. Other Generating Functions

In this section, we study some other generating functions.Theorem 21.
Ifα∈ℝ and n,j,q∈ℤ+, then
(98)∑j=0∞n!Aαq,nxAαq,nytnq+αn=1−t−q−αexp−xt1−texpx−yt1−tF0q−−;q+αq,q+1+αq,⋯,2q+α−1q;xytq1−tq.Proof.
Consider the series(99)∑n=0∞n!Aαq,nxAαq,nytnq+αn=∑n=0∞∑j=0nqn!yqjAαq,nx−1qjtnqj!n−qj!q+αqj.
By using Lemma3, we get
(100)∑n=0∞n!Aαq,nxAαq,nytnq+αn=∑n=0∞∑j=0∞n+qj!yqjAαq,n+qjx−1qjtn+qjqj!n!q+αqj,(101)=∑j=0∞∑n=0∞n+qj!Aαq,n+qjxtnqj!n!−ytqjq+αqj.
By using Theorem (92), we get
(102)∑j=0∞n!Aαq,nxAαq,nytnq+αn=∑j=0∞1−t−q−α−qjexp−xt1−tAαq,jx1−t−ytqjq+αqj=1−t−q−αexp−xt1−t∑j=0∞1−t−qjAαq,jx1−t−ytqjq+αqj=1−t−q−αexp−xt1−t×∑j=0∞Aαq,jx1−t−yt/1−tqjqqjq+α/qjq+1+α/qj⋯2q+α−1/qj=1−t−q−αexp−xt1−t∑j=0∞Aαq,jx1−t−yt/q1−tqjq+α/qjq+1+α/qj⋯2q+α−1/qj.
By using Equation (21), we get
(103)∑j=0∞n!Aαq,nxAαq,nytnq+αn=1−t−q−αexp−xt1−texpx−yt1−tF0q−−;q+αq,q+1+αq,⋯,2q+α−1q;xytq1−tq.Theorem 22.
Ift<1, α∈ℝ and c,n∈ℤ+, then
(104)1−t−1−αexpx1−t1−yt1−t−cFqqcq,c+1q,⋯,c+q−1q;−x/1−tyt/1−t1−yt/1−tqq+αq,q+1+αq,⋯,2q+α−1q;=∑n=0∞F2qq−nq,−n+1q,⋯,−n+q−1q,cq,c+1q,⋯,c+q−1q;q+αq,q+1+αq,⋯,2q+α−1q;qyqAαq,nxtn.Proof.
Consider the series(105)∑j=0∞n!Aαq,nxAαq,nytnq+αn=1−t−q−αexp−x+yt1−t×F0q−−;q+αq,q+1+αq,⋯2q+α−1q;xytq1−tq.
Applying Equation (92), we get
(106)ex1−tcFqqcq,c+1q,⋯,c+q−1q;−xt1−tqq+αq,q+1+αq,⋯,2q+α−1q;=∑j=0∞cqjAαq,qjxtqjq+αqj.
Replacingx by x1−t−1 and t by yt1−t−1 yields
(107)expx1−t1−yt1−t−cFqqcq,c+1q,⋯,c+q−1q;−x/1−tyt/1−t1−yt/1−tqq+αq,q+1+αq,⋯,2q+α−1q;=∑j=0∞cqjAαq,qjx/1−tyt/1−tqjq+αqj,multiplying both sides by 1−t−q−1exp−xt/1−t(108)1−t−q−αexpx1−texp−xt1−t1−yt1−t−c×Fqqcq,c+1q,⋯,c+q−1q;−x/1−tyt/1−t1−yt/1−tqq+αq,q+1+αq,⋯,2q+α−1q;=1−t−q−αexp−xt1−t2∑j=0∞cqjAαq,qjx/1−tyt/1−tqjq+αqj=∑j=0∞cqjAαq,qjx/1−t1−t−q−α−qjexp−xt/1−tqyqjtqjq+αqj.
By using Lemma4, we acquire
(109)1−t−1−αexpx1−t1−yt1−t−cFqqcq,c+1q,⋯,c+q−1q;−x/1−tyt/1−t1−yt/1−tqq+αq,q+1+αq,⋯,2q+α−1q;=∑n=0∞∑j=0nqcqjn!Aαq,qjxtn−qjyqjtqjqj!n−qj!q+αqj=∑n=0∞∑j=0nqcqj−nqjAαq,qjxtnyqjqj!q+αqj.
By using Lemma1 and 2, we get our required result.


## 10. Expansion of Polynomials

SinceAq,nax forms an orthogonal set, the classical technique for expanding a polynomial. As usual, we prefer to treat the problem by obtaining first the expansion of xqn and then using generating function techniques.Theorem 23.
Ifα∈ℝ and n,j∈ℤ+, then
(110)xqn=e−x∑j=0nqn!q+αnAαq,qjxn−qj!q+αqj.Proof.
Equation (21) then yields
(111)F0q−−;q+αq,q+1+αq,⋯2q+α−1q;−xtqq=e−x−t∑n=0∞Aαq,nxtnq+αn,(112)∑n=0∞−xt/qqnq+α/qnq+1+α/qn⋯2q+α−1/qnqn!=e−x∑n=0∞−1ntnn!∑n=0∞Aαq,nxtnq+αn,(113)∑n=0∞−xtqnq+αqnqn!=e−x∑n=0∞∑j=0∞−1ntnAαq,qjxtqjn!q+αqj.
By using Lemma4, we get
(114)∑n=0∞−1nxntnq+αnn!=e−x∑n=0∞∑j=0nq−1nAαq,qjxtnn−qj!q+αqj.
By equating the coefficient oftn, we get
(115)xqn=e−x∑j=0nqn!q+αnAαq,qjxn−qj!q+αqj.


## 11. Conclusion

Finally, in conclusion, we compromised the extended Laguerre polynomialsAαq,nx based on the Fqq,q>2. We obtained generating functions, recurrence relations, and Rodrigue’s formula for these extended Laguerre polynomials. In future work, we can extend it and can get more results. We will apply Laplace transformation, and Elzaki transformation and the same more transformations can apply on the results of extended Laguerre polynomials.



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*Source: 7326760-2022-04-15.xml*