# Q-Differential Equations of Higher Order and Structural Properties of the Approximate Roots for q-Modified Derangements’ Polynomials

Authors: Cheon Seoung Ryoo; Jung Yoog Kang
Journal: Journal of Mathematics
 (2022)

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## Abstract

The objective of this study is to findq-differential equations of higher order related to q-modified derangements’ polynomials and confirm the structure of approximation roots. Furthermore, it states several symmetric properties of q-differential equations of higher order and the special properties of the approximation roots of q-modified derangements’ polynomials.


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## Body

## 1. Introduction

### 1.1. Bernoulli Differential Equation

(1)dydx+pxy−gxym=0,is an equation, where m is any real number, and px and gx are continuous functions on the interval. Amongst all of the differential equations, the Bernoulli differential equation converts nonlinear equations into linear equations. If m=0 or m=1, the above equation is linear, and if not, the equation is nonlinear. By substituting u=y1−m, the Bernoulli differential equation can be reduced to a linear differential equation. Then, it concludes to a linear equation du/dx+1−mpxu=1−mgx for u. This equation can be applied to problems related to nonlinear differential equations, equations about the population expressed in logistic equations, or Verhulst equations, physics, etc.Ifm=0 in (1), the Bernoulli differential equation,(2)ddxDnx−Dnx+∑k=0nnk−1n−kxk=0,has a solution which is the generating function of modified derangements’ polynomials, see [1, 2].Fort≠1, the modified derangements’ numbers and polynomials can be expressed as(3)∑n=0∞Dntnn!=e−t1−t,∑n=0∞Dnxtnn!=e−t1−tetx,respectively.Table1 is the first few examples of the modified derangements’ numbers Dn and polynomials Dnx.Table 1
Examples ofDn and Dnx.
nDnDnx01110x21/21/2+1/2x231/31/3+1/2x+1/6x343/83/8+1/3x+1/4x2+1/24x4511/3011/30+3/6x+1/6x2+1/12x3+1/120x5⋮⋮⋮Based on the above concept andq-numbers, we consider the q-Bernoulli differential equation of the first order Dqy+pxy−gxym=0. In addition, it is possible to assume that the q-modified derangements’ polynomials are a solution of the following q-differential equation of the first order when m=0 in (1).The aim of this study is to find theq-Bernoulli differential equation with the solution of q-modified derangements’ polynomials. Also, we derive characteristic properties by visualizing the approximation roots of q-modified derangements’ polynomials.Several mathematicians discoveredq-differential equations by using special polynomials as a solution and studying their properties and identities, see [3–7]. The q-differential equation based on q-Hermit polynomials were studied by Hermoso, Huertas, and Lastra in [8]. In [9, 10], phenomena of roots for various kinds of polynomials related to differential equations were researched by Ryoo. Furthermore, various properties of polynomials using q-series, q-derivative, q-distribution, and so on were found, see [9, 11–13].To lay out the foundation for achieving the goal of this study, the following summarizes the definitions and theorems and makes arrangements.Jackson introduced theq-number, which plays an important role in q-calculus, see [4, 14]. Based on the discovery of q-number, useful results are studied in q-series, q-special functions, quantum algebras, q-discrete distribution, q-differential equations, q-calculus, etc., see [15, 16]. Here, we briefly review several concepts of q-calculus which we need for this study.Letn,q∈ℝ with q≠1. The number(4)nq=1−qn1−q,is called q-number, see [6, 16]. We note that limq⟶1nq=n. In particular, for k∈ℤ, kq is called q-integer.Theq-Gaussian binomial coefficients are defined by(5)mrq=mq!m−rq!rq!,where m and r are nonnegative integers, see [17]. For r=0, the value is 1 since the numerator and the denominator are both empty products. One notes nq!=nqn−1q…2q1q and 0q!=1.Definition 1.
Letz be any complex number with z<1. The q-exponential functions can be expressed as follows:(i)
eqz=∑n=0∞zn/nq!=∑n=0∞zn1−qn/q;qn=∑n=0∞zn1−qn/1−qn1−qn−1…1−q=∏k=0∞1−1−qqkz−1(ii)
For−1<q<1, Eqz=∑n=0∞qn2zn/nq!=∑n=0∞qn2−zn/q;qn=1+1−qxq∞=∏k=0∞1+1−qqkzwhere q;qn=1−qn1−qn−1…1−q is the q-Pochhammer symbol.
We note thatlimq⟶1eqz=ez, see [12, 16].Theorem 1.
From Definition1, we note that(6)ieqxeqy=eqx+y,ifyx=qxy,iieqxEq−x=1,iiieq−1x=Eqx.Definition 2.
Theq-derivative of a function f with respect to x is defined by(7)Dq,xfx≔Dqfx=fx−fqx1−qx,forx≠0,and Dqf0=f′0.
We can prove thatf is differentiable at zero, and it is clear that Dqxn=nqxn−1, see [6, 14–16]. From Definition 2, there are some formulae for q-derivative.Theorem 2.
From Definition2, we note that(8)iDqfxgx=qxDqfx+fqxDqgx=fxDqgx+gqxDqfx,iiDqfxgx=gqxDqfx−fqxDqgxgxgqx=gxDqfx−fxDqgxgxgqx,iiifor any constantsaandb,Dqafx+bgx=aDqfx+bDqgx.Our ultimate purpose is to find the solution ofq-modified derangements’ polynomials by observing various q-differential equations of higher order. In Section 2, we define the q-modified derangements’ numbers and polynomials, mention several forms of q-differential equations of higher order, and check its associated symmetric properties. Lastly, in Section 3, by observing the values of q-modified derangements’ numbers, the approximation roots of q-modified derangements’ polynomials will be shown, and several conjectures for those numbers and polynomials will be organized.


## 1.1. Bernoulli Differential Equation

(1)dydx+pxy−gxym=0,is an equation, where m is any real number, and px and gx are continuous functions on the interval. Amongst all of the differential equations, the Bernoulli differential equation converts nonlinear equations into linear equations. If m=0 or m=1, the above equation is linear, and if not, the equation is nonlinear. By substituting u=y1−m, the Bernoulli differential equation can be reduced to a linear differential equation. Then, it concludes to a linear equation du/dx+1−mpxu=1−mgx for u. This equation can be applied to problems related to nonlinear differential equations, equations about the population expressed in logistic equations, or Verhulst equations, physics, etc.Ifm=0 in (1), the Bernoulli differential equation,(2)ddxDnx−Dnx+∑k=0nnk−1n−kxk=0,has a solution which is the generating function of modified derangements’ polynomials, see [1, 2].Fort≠1, the modified derangements’ numbers and polynomials can be expressed as(3)∑n=0∞Dntnn!=e−t1−t,∑n=0∞Dnxtnn!=e−t1−tetx,respectively.Table1 is the first few examples of the modified derangements’ numbers Dn and polynomials Dnx.Table 1
Examples ofDn and Dnx.
nDnDnx01110x21/21/2+1/2x231/31/3+1/2x+1/6x343/83/8+1/3x+1/4x2+1/24x4511/3011/30+3/6x+1/6x2+1/12x3+1/120x5⋮⋮⋮Based on the above concept andq-numbers, we consider the q-Bernoulli differential equation of the first order Dqy+pxy−gxym=0. In addition, it is possible to assume that the q-modified derangements’ polynomials are a solution of the following q-differential equation of the first order when m=0 in (1).The aim of this study is to find theq-Bernoulli differential equation with the solution of q-modified derangements’ polynomials. Also, we derive characteristic properties by visualizing the approximation roots of q-modified derangements’ polynomials.Several mathematicians discoveredq-differential equations by using special polynomials as a solution and studying their properties and identities, see [3–7]. The q-differential equation based on q-Hermit polynomials were studied by Hermoso, Huertas, and Lastra in [8]. In [9, 10], phenomena of roots for various kinds of polynomials related to differential equations were researched by Ryoo. Furthermore, various properties of polynomials using q-series, q-derivative, q-distribution, and so on were found, see [9, 11–13].To lay out the foundation for achieving the goal of this study, the following summarizes the definitions and theorems and makes arrangements.Jackson introduced theq-number, which plays an important role in q-calculus, see [4, 14]. Based on the discovery of q-number, useful results are studied in q-series, q-special functions, quantum algebras, q-discrete distribution, q-differential equations, q-calculus, etc., see [15, 16]. Here, we briefly review several concepts of q-calculus which we need for this study.Letn,q∈ℝ with q≠1. The number(4)nq=1−qn1−q,is called q-number, see [6, 16]. We note that limq⟶1nq=n. In particular, for k∈ℤ, kq is called q-integer.Theq-Gaussian binomial coefficients are defined by(5)mrq=mq!m−rq!rq!,where m and r are nonnegative integers, see [17]. For r=0, the value is 1 since the numerator and the denominator are both empty products. One notes nq!=nqn−1q…2q1q and 0q!=1.Definition 1.
Letz be any complex number with z<1. The q-exponential functions can be expressed as follows:(i)
eqz=∑n=0∞zn/nq!=∑n=0∞zn1−qn/q;qn=∑n=0∞zn1−qn/1−qn1−qn−1…1−q=∏k=0∞1−1−qqkz−1(ii)
For−1<q<1, Eqz=∑n=0∞qn2zn/nq!=∑n=0∞qn2−zn/q;qn=1+1−qxq∞=∏k=0∞1+1−qqkzwhere q;qn=1−qn1−qn−1…1−q is the q-Pochhammer symbol.
We note thatlimq⟶1eqz=ez, see [12, 16].Theorem 1.
From Definition1, we note that(6)ieqxeqy=eqx+y,ifyx=qxy,iieqxEq−x=1,iiieq−1x=Eqx.Definition 2.
Theq-derivative of a function f with respect to x is defined by(7)Dq,xfx≔Dqfx=fx−fqx1−qx,forx≠0,and Dqf0=f′0.
We can prove thatf is differentiable at zero, and it is clear that Dqxn=nqxn−1, see [6, 14–16]. From Definition 2, there are some formulae for q-derivative.Theorem 2.
From Definition2, we note that(8)iDqfxgx=qxDqfx+fqxDqgx=fxDqgx+gqxDqfx,iiDqfxgx=gqxDqfx−fqxDqgxgxgqx=gxDqfx−fxDqgxgxgqx,iiifor any constantsaandb,Dqafx+bgx=aDqfx+bDqgx.Our ultimate purpose is to find the solution ofq-modified derangements’ polynomials by observing various q-differential equations of higher order. In Section 2, we define the q-modified derangements’ numbers and polynomials, mention several forms of q-differential equations of higher order, and check its associated symmetric properties. Lastly, in Section 3, by observing the values of q-modified derangements’ numbers, the approximation roots of q-modified derangements’ polynomials will be shown, and several conjectures for those numbers and polynomials will be organized.


## 2. Various Types ofq-Differential Equations of Higher Order Associated with the q-Modified Derangements’ Polynomials

In this section,q-modified derangements’ polynomials are defined, and various kinds of q-differential equation of higher order associated with these polynomials are introduced. Moreover, we find several symmetric properties of q-differential equation of higher order.Definition 3.
Fort≠1 and q<1, the q-modified derangements’ polynomials Dn,qx is defined with the following generating function as follows:(9)∑n=0∞Dn,qxtnnq!=eq−t1−teqtx.
From Definition3, we note that(10)∑n=0∞Dn,qtnnq!=eq−t1−t.
Here, we defineDn,q as the q-modified derangements’ numbers. The q-modified derangements’ numbers Dn,q has the relation Dn,q=Dn,q1 with the polynomials of q-derangements, see [1].Theorem 3.
Theq-modified derangements’ polynomials Dn,qx is a solution of the q-differential equation which can be given as(11)Dq,x1Dn,qx−Dn,qx+∑k=0nnkq−1n−kxk=0.Proof.
Whent≠1, the generating function of the q-modified derangements’ polynomials have(12)1−t∑n=0∞Dn,qxtnnq!=∑n=0∞∑k=0nnkq−1n−kxktnnq!.
The left-hand side of (12) can be changed to(13)1−t∑n=0∞Dn,qxtnnq!=∑n=0∞Dn,qx−nqDn−1,qxtnnq!.
Using (12) and (13), applying the Cauchy product rule and comparing the coefficients of both sides of result equation, we have(14)Dn,qx−nqDn−1,qx=∑k=0nnkq−1n−kxk.
Applyingq-derivative in Dn,qx, we find a relation between Dn−1,qx and Dq,x1Dn,qx as(15)Dn−1,qx=1nqDq,x1Dn,qx.
SubstitutingDq,x1Dn,qx of (15) to the left-hand side of (14), we achieve the shown result.Corollary 1.
Whenq⟶1 in Theorem 3, one holds that(16)ddxDnx−Dnx+∑k=0nnk−1n−kxk=0,where Dnx is the modified derangements’ polynomials.Theorem 4.
The solution of the followingq-differential equation of higher order is the q-modified derangements’ polynomials Dn,qx which can be given as follows:(17)qn2−nqqn−12nq!Dq,xnDn,qx+qn−12−n−1qqn−22n−1q!Dq,xn−1Dn,qx+qn−22−n−2qqn−32n−2q!Dq,xn−2Dn,qx+…+q32−3qq3q!Dq,x3Dn,qx+q−2q2q!Dq,x2Dn,qx+Dn,qx−xn=0.Proof.
Using a property ofq-exponential function eqtEq−t=1 in the generating function of the q-modified derangements’ polynomials, we find(18)∑n=0∞Dn,qx=eq−t1−teqtx,=11−tEqteqtx.
Suppose that1−tEqt≠0 in (18). Then, we have(19)1−tEqt∑n=0∞Dn,qx=∑n=0∞xntnnq!.
From the power series ofq-exponential functions, the left-hand side of (19) can be transformed as(20)1−tEqt∑n=0∞Dn,qx=∑n=0∞qn2−nqqn−12tnnq!∑n=0∞Dn,qxtnnq!,=∑n=0∞∑k=0nnkqqk2−kqqk−12Dn−k,qxtnnq!.
Comparing (19) and (20), we have(21)∑k=0nnkqqk2−kqqk−12Dn−k,qx=xn.
Applyingq-derivative in Dn,qx, we obtain a relation such as(22)Dn−k,qx=n−kq!nq!Dq,xkDn,qx.
From (21) and (22), we find (23)∑k=0nnkqqk2−kqqk−12Dn−k,qx=∑k=0nqk2−kqqk−12kq!Dq,xkDn,qx.
Combining (21) and (23), we complete the desired result.Corollary 2.
Letq⟶1 in Theorem 4. Then, we obtain(24)1−nn!dndxnDnx+2−nn−1!dn−1dxn−1Dnx+3−nn−2!dn−2dxn−2Dnx+…+23!d3dx3Dnx+12!d2dx2Dnx+Dnx−xn=0,where Dnx is the modified derangements’ polynomials.Theorem 5.
Theq-modified derangements’ polynomials Dn,qx is a solution of the following q-differential equation of higher order which is combined with the q-modified derangements’ numbers:(25)∑k=0n−1n−1kq−1k−1+Dk,qqn−k−22n−1q!Dq,xn−1Dn−1,qx+∑k=0n−2n−2kq−1k−1+Dk,qqn−k−32+1n−2q!Dq,xn−2Dn−1,qx+⋯+∑k=022kq−1k−1+Dk,qq1−k2+n−32q!Dq,x2Dn−1,qx+∑k=011kq−1k−1+Dk,qqn−2Dq,x1Dn−1,qx−1−D0,q−qxqn−1Dn−1,qx−Dn,qqx=0.Proof.
Initially, we apply theq-derivative after substituting qx instead of x in the generating function of the q-modified derangements’ polynomials Dn,qx. Then, we obtain(26)Dq,t∑n=0∞Dn,qqxtnnq!=eqqtxeq−t−1−teq−t1−t1−qt+qxeq−qt1−qteqqtx,=∑n=0∞qnDn,qxtnnq!eq−t1−teq−qt−eq−teq−qt+qx,=∑n=0∞qnDn,qxtnnq!Eqqt∑n=0∞Dn,qtnnq!−Eqqt∑n=0∞−1ntnnq!+qx,=∑n=0∞∑l=0n∑k=0lnlqlkq−1k−1+Dk,qql−k+12+n−lDn−l,qx+qn+1xDn,qxtnnq!.
From (26), we have(27)tDq,t∑n=0∞Dn,qqxtnnq!=∑n=0∞nq∑l=0n−1∑k=0ln−1lqlkq−1k−1+Dk,qql−k+12+n−l−1Dn−l−1,qxtnnq!+∑n=0∞nqqnxDn−1,qxtnnq!.
From the generating function of theq-modified derangements’ polynomials Dn,qx, we also have(28)tDq,t∑n=0∞Dn,qqxtnnq!=∑n=0∞nqDn,qqxtnnq!.
Comparing (27) and (28), we obtain(29)Dn,qqx−qnxDn−1,qx=∑l=0n−1∑k=0ln−1lqlkq−1k−1+Dk,qql−k+12+n−l−1Dn−l−1,qx.
Applying relation (22) in (29), we derive(30)Dn,qqx−qnxDn−1,qx=∑l=0n−1∑k=0llkq−1k−1+Dk,qql−k−12+n−l−1lq!Dq,xlDn−1,qx,and this equation gives us the required result.□Corollary 3.
Settingq⟶1 in Theorem 5, we obtain(31)∑k=0n−1n−1k−1k−1+Dkn−1!dn−1dxn−1Dn−1x+∑k=0n−2n−2k−1k−1+Dkn−2!dn−2dxn−2Dn−1x+⋯+∑k=022k−1k−1+Dk2!d2dx2Dn−1x+∑k=011k−1k−1+DkddxDn−1x−1−D0−xDn−1x−Dnx=0,where Dn is the modified derangements’ number and Dnx is the modified derangements’ polynomials.Theorem 6.
Theq-modified derangements’ polynomials Dn,qx is a solution of the q-differential equation of higher order which is(32)∑k=0n−1n−1kq−1k+1qn−k−12+n−k−1+qk2+kDn−k−1,qqn−1n−1q!Dq,xn−1Dn−1,qq−1x+∑k=0n−2n−2kq−1k+1qn−k−22+n−k−2+qk2+kDn−k−2,qqn−1n−2q!Dq,xn−2Dn−1,qq−1x+⋯+∑k=022kq−1k+1q2−k2+2−k+qk2+kD2−k,qqn−12q!Dq,x2Dn−1,qq−1x+∑k=011kq−1k+1q1−k2+1−k+qk2+kD1−k,qqn−1Dq,x1Dn−1,qq−1x+1+x+D0,qqn−1Dn−1,qq−1x−Dn,qx=0,where Dn,q is the q-modified derangements’ numbers.Proof.
apply theq-derivative after substituting qt instead of t in Dn,qx. Then, we have(33)Dq,t∑n=0∞qnDn,qxtnnq!=eqqtxqeq−qt−q1−qteq−qt1−qt1−q2t+qxeq−q2t1−q2teqqtx,=∑n=0∞q2n+1Dn,qq−1xtnnq!Eqq2t∑n=0∞qnDn,qtnnq!−Eqq2teq−qt+x,=∑n=0∞∑l=0n∑k=0lnlqlkq−1k+1ql−k2+l−k+qk2+kDl−k,qq2n−l+1Dn−l,qq−1xtnnq!+∑n=0∞q2n+1xDn,qq−1xtnnq!.
In a similar way to find (27) and (28), we obtain(34)tDq,t∑n=0∞qnDn,qxtnnq!=∑n=0∞nq∑l=0n−1∑k=0ln−1lqlkq−1k+1ql−k2+l−k+qk2+kDl−k,qq2n−l−1Dn−l−1,qq−1xtnnq!+∑n=0∞nqq2n−1xDn−1,qq−1xtnnq!,(35)tDq,t∑n=0∞qnDn,qxtnnq!=∑n=0∞nqqnDn,qxtnnq!.
From (34) and (35), we have(36)∑l=0n−1∑k=0ln−1lqlkq−1k+1ql−k2+l−k+qk2+kDl−k,qqn−l−1Dn−l−1,qq−1x=Dn,qx−qn−1xDn−1,qq−1x.
Here, we find a relation betweenDn,qq−1x and Dq,xkDn,qq−1x as follows:(37)Dn−k,qq−1x=qkn−kq!nq!Dq,xkDn,qq−1x.
Using (37) in the left-hand side of (36), we have(38)∑l=0n−1∑k=0llkq−1k+1ql−k2+l−k+qk2+kDl−k,qqn−1lq!Dq,xlDn−1,qq−1x=Dn,qx−qn−1xDn−1,qq−1x.
By (38), it is possible to finish the proof of Theorem 6.Corollary 4.
Puttingq⟶1 in Theorem 6, one holds that(39)∑k=0n−1n−1k−1k+1+Dn−k−1n−1!dn−1dxn−1Dn−1x+∑k=0n−2n−2k−1k+1+Dn−k−2n−2!dn−2dxn−2Dn−1x+⋯+∑k=022k−1k+1+D2−k2!d2dx2Dn−1,qx+∑k=011k−1k+1+D1−kddxDn−1x+1+x+D0Dn−1x−Dnx=0,where Dn is the modified derangements’ numbers and Dnx is the modified derangements’ polynomials.From now on, we take several suitable forms to find symmetric properties ofq-differential equation of higher order. To obtain various symmetric properties of q-differential equation of higher order which combines other numbers and polynomials, the below forms A,B are the basic forms.Theorem 7.
Leta≠0, b≠0, and 0<q<1. Then, we obtain(40)bnanDn,qnq!Dq,xnDn,qb−1x+bnan−1Dn−1,qn−1q!Dq,xn−1Dn,qb−1x+⋯+bna2D2,q2q!Dq,x2Dn,qb−1x+bnaD1,qDq,x1Dn,qb−1x+bnD0,qDn,qb−1x=anbnDn,qnq!Dq,xnDn,qa−1x+anbn−1Dn−1,qn−1q!Dq,xn−1Dn,qa−1x+⋯+anb2D2,q2q!Dq,x2Dn,qa−1x+anbD1,qDq,x1Dn,qa−1x+anD0,qDn,qa−1x.Proof.
To obtain a symmetric property ofq-differential equation of higher order for q-modified derangements’ polynomials Dn,qx, we consider a form A as(41)A:=eq−ateq−bteqtx1−at1−bt.
From the formA, we have(42)eq−at1−ateq−bt1−bteqtx=∑n=0∞∑k=0nnkqakbn−kDk,qDn−k,qb−1xtnnq!,(43)eq−bt1−bteq−at1−ateqtx=∑n=0∞∑k=0nnkqbkan−kDk,qDn−k,qa−1xtnnq!.
From (42) and (43), we find(44)∑k=0nnkqakbn−kDk,qDn−k,qb−1x=∑k=0nnkqbkan−kDk,qDn−k,qa−1x.
Replacing (37) to (44), we obtain(45)bn∑k=0nakDk,qkq!Dq,xkDn,qb−1x=an∑k=0nbkDk,qkq!Dq,xkDn,qa−1x.
From (45), we complete the proof of Theorem 7.Corollary 5.
Settinga=1 in Theorem 7, we have(46)bnDn,qnq!Dq,xnDn,qb−1x+bnDn−1,qn−1q!Dq,xn−1Dn,qb−1x+⋯+bnD2,q2q!Dq,x2Dn,qb−1x+bnD1,qDq,x1Dn,qb−1x+bnD0,qDn,qb−1x=bnDn,qnq!Dq,xnDn,qx+bn−1Dn−1,qn−1q!Dq,xn−1Dn,qx+⋯+b2D2,q2q!Dq,x2Dn,qx+bD1,qDq,x1Dn,qx+D0,qDn,qx.Corollary 6.
considerq⟶1 in Theorem 7; then, we get(47)bnanDnn!dndxnDnb−1x+bnan−1Dn−1n−1!dn−1dxn−1Dnb−1x+⋯+bna2D22!d2dx2Dnb−1x+bnaD1ddxDnb−1x+bnD0Dnb−1x=anbnDnn!dndxnDna−1x+anbn−1Dn−1n−1!dn−1dxn−1Dna−1x+⋯+anb2D22!d2dx2Dna−1x+anbD1ddxDna−1x+anD0Dna−1x.Theorem 8.
Leta≠0, b≠0, and 0<q<1. Then, we have(48)bnanDn,qa−1xnq!Dq,ynDn,qb−1y+bnan−1Dn−1,qa−1xn−1q!Dq,yn−1Dn,qb−1y+⋯+bnaD1,qa−1xDq,y1Dn,qb−1y+bnD0,qa−1xDn,qb−1y=anbnDn,qb−1ynq!Dq,ynDn,qa−1y+anbn−1Dn−1,qb−1xn−1q!Dq,yn−1Dn,qa−1y+⋯+anbD1,qb−1xDq,y1Dn,qa−1y+anD0,qb−1xDn,qa−1y.Proof.
Suppose formB as(49)B:=eq−ateq−bteqtxeqty1−at1−bt.
UsingB, the desired result can be obtained in a similar way to the proof of Theorem 7. Therefore, we omit the proof process.Corollary 7.
Settinga=1 in Theorem 8, we have(50)bnDn,qxnq!Dq,ynDn,qb−1y+bnDn−1,qxn−1q!Dq,yn−1Dn,qb−1y+⋯+bnD1,qxDq,y1Dn,qb−1y+bnD0,qxDn,qb−1y=bnDn,qb−1xnq!Dq,ynDn,qy+bn−1Dn−1,qb−1xn−1q!Dq,yn−1Dn,qy+⋯+bD1,qb−1xDq,y1Dn,qy+D0,qb−1xDn,qy.Corollary 8.
considerq⟶1 in Theorem 8; then, we get(51)bnanDna−1xn!dndynDnb−1y+bnan−1Dn−1a−1xn−1!dn−1dyn−1Dnb−1y+⋯+bnaD1a−1xddyDnb−1y+bnD0a−1xDnb−1y=anbnDnb−1xn!dndynDna−1y+anbn−1Dn−1b−1xn−1!dn−1dyn−1Dna−1y+⋯+anbD1b−1xddyDna−1y+anD0b−1xDna−1y.


## 3. Visualization of the Approximation Roots forq-Modified Derangements’ Polynomials

This section handles the approximation roots of theq-modified derangements’ polynomial. To confirm several conjectures for q-modified derangements, we show the structure of the approximation roots of those polynomials. We use MATHEMATICA in order to obtain the pictures and the results of calculation.Based on the generating function ofq-modified derangements’ numbers, Dn,q are found as follows:(52)D0,q=1,D1,q=0,D2,q=1,D3,q=q1+q,D4,q=1+q+q21+q2+q3,D5,q=q1+q1+q22+2q+2q2+2q3+2q4+q5,D6,q=1+q+q2+q3+q4×1+q+4q2+6q3+8q4+9q5+8q6+7q7+5q8+3q9+q10,….From the aboveq-modified derangements’ numbers, Table 2 shows the approximation values of Dn,q which appear with changes in values of q. In Table 1, as the value of q increases, we can observe that the approximation value of q-modified derangements’ numbers also increases.Table 2
Approximate values ofDn,q.
nq0.10.40.60.9901111100002111130.110.560.961.970141.122211.909443.088968.7629850.2468882.149816.1219141.947361.274324.5683415.5907246.47570.415916.6014336.88571673.4281.4621211.995291.665312929.1⋮⋮⋮⋮⋮By using Table2, we can see the location of Dn,q shown by varying n and q, as shown in Figure 1. The nonnegative integers of the x-axis represent the values of n in Figure 1. Here, the lines display variations of the approximation values for q-modified derangements’ numbers. The blue dots, the yellow squares, and the green rhombuses in Figure 1 are the approximation values of q-modified derangements’ numbers when q=0.6,0.4,0.1, respectively. For example, the blue dot above the value 6 on the real axis shows the approximation value of D6,0.6, and the yellow square indicates the approximation value of D6,0.4.Figure 1
Locations ofDn,q for 0≤n≤8 and q=0.6,0.4,0.1.From Table2 and Figure 1, we can find the following conjecture.Conjecture 1.
Asn increases and q approaches 1, the approximation value of Dn,q increases.Using the generating function ofq-modified derangements’ polynomials, Dn,qx are found as follows:(53)D0,qx=1,D1,qx=x,D2,qx=1+x2,D3,qx=2q+2q2+q3+1+q+q2x+q+q2+q3x2+x3,….Based on the above polynomial, we find special properties of approximations values that appear depending on the value ofq of the q-modified derangements’ polynomials. Here, we set the values of q to 0.01, 0.001, and 0.0001, respectively, because the extremely small value of q represents the properties of the q-modified derangements’ polynomials. We experiment with the q-modified derangements’ polynomials for two purposes. The first is to check the structure of the approximation roots and find the polygon that is most similar to it. The second is to find the approximation values of the real number among the approximation roots appearing in the q-modified derangements’ polynomials.To obtain the first purpose, Figure2 shows the structure of the approximate roots of q-modified derangements’ polynomials. Here, the range of n is from 0 to 50, and the value of q changes. In Figure 2, the value of q in (a) is 0.01, the value of q in (b) is 0.001, and the value of q in (c) is 0.0001. A characteristic that can be confirmed in the structure of the approximation roots in Figure 2(c) is that, as n increases, one approximation root close to the origin continues to accumulate. The structures of the approximation roots shown in Figure 2 seem to become more circular as n increases.Based on the results of Figure2, we find Figure 3. Figure 3 shows the structure of approximation roots when the value of n is 50. In Figure 3, the value of q in (a) is 0.01, the value of q in (b) is 0.001, and the value of q in (c) is 0.0001. Here, we set the red dots to indicate the approximation roots, the blue dots to indicate the center of the circle, and the blue lines to connect the red dots. Also, to obtain the blue lines closest to the approximation roots, the real roots are excluded.Table3 is the result of calculating the exact values in Figure 3. From Table 3, we can see that, as the value of q gets smaller, the structure of the approximation roots has a shape closer to a circle, and the radius of the circle gets closer to 1.From Figures2 and 3 and Table 3, we make the following conjecture.Figure 2
Locations ofDn,qx for 0≤n≤50: (a) q=0.01, (b) q=0.001, and (c) q=0.0001.
(a)(b)(c)Figure 3
Approximation roots and circles ofD50,qx: (a) q=0.01, (b) q=0.001, and (c) q=0.0001.
(a)(b)(c)Table 3
The circle of approximation roots ofD50,qx.
The centerx,yThe radiusThe error rangeq=0.010.0202672,1.19233×10−121.262170.0319478q=0.0010.0108304,−2.93695×10−111.015930.000242633q=0.00010.00120143,4.84225×10−141.001243.23829×10−8Conjecture 2.
Asq becomes smaller than 0.001, the structure of the approximation roots of D50,qx located on a circle will have a radius close to 1.From now on, based on Figure2, we find approximation roots by dividing them into real and imaginary roots. Since our concern is the real roots, it can be obtained as shown in Table 4 by finding and expressing the real roots.We can find a special property forq-modified derangements’ polynomials in Table 4. Table 4 shows that only odd-order polynomials of Dn,qx have approximation real roots even if the value of q changes, and only one approximation real root exists. Therefore, we can make the following assumptions.Table 4
Approximation real roots ofDn,qx.
nq0.010.0010.0001⋮⋮⋮⋮5−0.0600175−0.00600092−0.0006000167−0.119695−0.0120024−0.0012000389−0.197899−0.0200032−0.0020000610⋮⋮⋮⋮19−0.718468−0.089792−0.0090001120⋮⋮⋮⋮291.00717−0.207049−0.020997830⋮⋮⋮⋮39−1.12658−0.36285−0.037983440⋮⋮⋮⋮Conjecture 3.
q-modified derangements’ polynomials have only one approximation real root in odd-order polynomials regardless of the change in the q-number.


## 4. Conclusion

We organized theq-differential equation with the solution of q-modified derangements’ polynomials and find the corresponding symmetric properties. We also looked at the distribution of approximation roots of q-modified derangements’ polynomials and their phenomena. As a result, we can make some assumptions and think that the related research should be continued.



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*Source: 2965684-2022-11-14.xml*